Math Nation - Geometry - Volume 2 of 2

Table of Contents Unit 8 - Proving Relationships and Theorems 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Lesson

Proving Properties of Parallelograms Using Properties of Parallelograms Special Parallelograms Kites and Parallelograms Trapezoids Solving Real-World Problems With Quadrilaterals Trapezoid Midsegment Theorem

Page Number 7 13 19 25 32 39 42

Unit 9 - Two- and Three-Dimensional Shapes 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

Lesson Finding the Area of Two-Dimensional Figures Density Based on Area Cross Sections Solids of Rotation Cavalieri’s Principle Finding Volume Volume in the Real World Surface Area of Three-Dimensional Figures Surface Area in the Real World Determining How Dilations Affect Area Determining Surface Area Given Dimension Changes

Page Number 51 59 65 70 76 82 88 93 100 104 112

Unit 10 - Arcs and Angle Relationships in Circles 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

Sectors Arc Length Radian Measure Solving Problems Solving Problems Solving Problems Solving Problems Solving Problems

Lesson

Involving Involving Involving Involving Involving

Arc Length and Area of a Sector Arc Measure and Central Angles Inscribed Angles Intersecting Chords Intersection of Secants

Page Number 119 124 130 135 141 148 153 159

i

10.9 Solving Problems Involving Tangents 10.H1 Constructing Tangent Lines

164 169

Unit 11 - Justifying Relationships in Polygons With Circles Lesson 11.1 Constructing a Circumscribed Circle of a Triangle 11.2 Constructing the Incenter of a Triangle 11.H1 Inscribing Polygons in Circles 11.3 Solving Problems Involving Triangles Inscribed in Circles 11.4 Solving Problems Involving Quadrilaterals Inscribed in Circles 11.5 Interior and Exterior Angles of Triangles 11.6 Triangle Inequality Theorem 11.7 Centroid of a Triangle 11.8 Triangle Midsegment Theorem 11.9 Solving Problems Using Triangle Relationships

Page Number 177 182 187 191 197 202 207 213 220 225

Unit 12 - Segment Relationships in Circles 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9

Lesson Solving Problems With Lengths of Intersecting Chords Solving Problems With Lengths of Two Secants Solving Problems With Secant and Tangent Segments Solving Problems With Lengths of Two Tangents Pythagorean Theorem and Tangent Lines Lengths of Secants, Tangents, Segments, and Chords in Circles Deriving the Equation of a Circle Standard Form of a Circle Graphing Circles

Page Number 231 237 242 246 251 257 260 266 272

Unit 13 - Shapes on a Coordinate Plane 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8

ii

Parallel and Perpendicular Lines Circles on the Coordinate Plane Determining Points on a Circle Partitioning Line Segments Weighted Averages Classifying Triangles Centroids of Triangles Classifying Quadrilaterals – Part 1

Lesson

Page Number 281 288 294 299 306 314 321 327

13.9 13.10 13.11 13.12 13.13 13.14

Classifying Quadrilaterals – Part 2 Classifying Quadrilaterals – Part 3 What Is the Figure?

Congruence and Similarity Finding Perimeter and Area on the Coordinate Plane Using Coordinate Geometry to Solve Real-World Problems

334 339 345 348 354 360

Additional Resources Resource

Page Number

Benchmark Index

366

Appendix C: 6-12 Mathematics Glossary

368

Appendix D: Properties of Operations, Equality and Inequality

375

Appendix E: K-12 Formulas

377

iii

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

iv

Unit 8 - Proving Relationships and Theorems 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Lesson Proving Properties of Parallelograms Using Properties of Parallelograms Special Parallelograms Kites and Parallelograms Trapezoids Solving Real-World Problems With Quadrilaterals Trapezoid Midsegment Theorem

Unit 8: Proving Relationships and Theorems

Page Number 7 13 19 25 32 39 42

5

The following B.E.S.T Standards will be covered in this unit: MA.912.GR.1.4 Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships and theorems of parallelograms.

• Clarification 1: Postulates, relationships and theorems include opposite sides are congruent, consecutive angles are supplementary, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and rectangles are parallelograms with congruent diagonals.

• Clarification 2: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs.

• Clarification 3: Instruction focuses on helping a student choose a method they can use reliably.

MA.912.GR.1.5 Prove relationships and theorems about trapezoids. Solve mathematical and real-world problems involving postulates, relationships and theorems of trapezoids.

• Clarification 1: Postulates, relationships and theorems include the Trapezoid Midsegment Theorem and for isosceles trapezoids: base angles are congruent, opposite angles are supplementary and diagonals are congruent.

• Clarification 2: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs.

• Clarification 3: Instruction focuses on helping a student choose a method they can use reliably.

MA.912.LT.4.10 Judge the validity of arguments and give counterexamples to disprove statements.

• Clarification 1: Within the Geometry course, instruction focuses on the connection to proofs within the course.

6

Unit 8: Proving Relationships and Theorems

Unit 8, Lesson 1: Proving Properties of Parallelograms Benchmarks MA.912.GR.1.4 Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships, and theorems of parallelograms.

MA.912.LT.4.10 Judge the validity of arguments and give counterexamples to disprove statements.

Learning Targets    

I can define a parallelogram.

I can prove opposite sides and opposite angles of a parallelogram are congruent.

I can prove consecutive angles of a parallelogram are supplementary.

I can prove diagonals of a parallelogram bisect each other.

8.1.1 Warm-Up: Cartographer Dave Imus is a cartographer whose maps have won many awards. Dave sees the world through understanding places.

Dave’s maps are considered art.

1.

Have you ever used a paper map?

2.

Since you can’t put everything on a map, what three places do you think should be on the map of Florida?

Unit 8: Proving Relationships and Theorems

7

8.1.2 Collaboration: Parallel Lines and Four-Sided Figures 1.

On a piece of patty paper, draw one set of parallel lines. Overlap your patty paper with a partner’s parallel lines to create an enclosed figure. Each partner should trace the enclosed shape onto a new piece of patty paper. a.

8.1.3 Exploration: Properties of Parallelograms

b.

Justify your classification.

c.

Ask two classmates how they classified the figure they created. Record their classification and write your summary of their justification. You are the only person who should write in the first two columns of the table. Have the person initial next to your summary stating that your summary was correct.

My Summary of Their Justification

Using a ruler, measure the pairs of opposite sides in each parallelogram you and your partner created. a.

What do you notice?

b.

Make a conjecture about the measures of opposite sides of a parallelogram.

Initials

2.

8

A quadrilateral containing two pairs of parallel sides is a parallelogram. A member of the following shape classes: polygons, quadrilaterals, trapezoids. Sub-classes include rectangles, rhombi, and squares.

1.

Classify the enclosed shape.

Classification

Using a ruler, draw the diagonals of one of the parallelograms. a.

Determine the lengths of each diagonal.

b.

Mark the point where the two diagonals intersect.

Unit 8: Proving Relationships and Theorems

c.

What do you notice about the point of intersection and its relationship to each diagonal?

b.

Make a conjecture about the measures of opposite angles of a parallelogram.

d.

Repeat the same steps to draw and measure the diagonals in your second parallelogram.

c.

Make a conjecture about the measures of consecutive angles of a parallelogram.

e.

Make a conjecture about the diagonals of a parallelogram.

4.

3.

Using a protractor, measure the pairs of opposite interior angles of your parallelograms. a.

What do you notice?

Based on your conjectures, select all of the statements about a parallelogram that are true.  A parallelogram has only one set of opposite parallel sides.  A parallelogram has two sets of opposite parallel sides.  A parallelogram has only one set of opposite congruent sides.  A parallelogram has two sets of opposite congruent sides.  Consecutive angles of a parallelogram are congruent.  Consecutive angles of a parallelogram are supplementary.  Opposite angles of a parallelogram are congruent.  Opposite angles of a parallelogram are supplementary.  Diagonals of a parallelogram bisect each other.

Unit 8: Proving Relationships and Theorems

9

5.

For each true statement in the previous problem, draw and name a parallelogram. Using the information in the statement, mark the diagram and write a mathematical statement. The first row has been completed.

Statement

A parallelogram has two sets of opposite parallel sides.

Drawing

6.

Given

̅​̅​̅​̅ 𝐴𝐴𝐴𝐴 ∥ ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵 ̅​̅​̅​̅ ̅​̅​̅​̅ 𝐴𝐴𝐴𝐴 ∥ 𝐷𝐷𝐷𝐷

The Venn diagram shows the properties of a parallelogram and another quadrilateral called a kite.

Discuss with your partner why a kite is not considered a parallelogram.

Parallelogram opposite sides ∥

opposite ∠s ≅

consecutive ∠s supplementary

diagonals bisect

opposite sides ≅

10

consecutive sides ≅ diagonals ⊥

one pair of opposite ∠s ≅

Theorems about Parallelograms • •

Kite

•

•

Opposite sides of a parallelogram are congruent.

Opposite angles of a parallelogram are congruent.

Consecutive interior angles of a parallelogram are supplementary.

The diagonals of a parallelogram bisect each other.

Unit 8: Proving Relationships and Theorems

Unit 8: Proving Relationships and Theorems

11

3.

Complete the following flowchart proof. Statements go in the boxes and reasons go below the boxes.

8.1.5 Wrap-Up: Cool Down 1.

Write down one fact you knew about parallelograms before the lesson.

2.

Write down one fact you learned about parallelograms during the lesson.

Given: is a parallelogram.

Prove: ∠𝑊𝑊 is supplementary to ∠𝐷𝐷, ∠𝐷𝐷 is supplementary to ∠𝐹𝐹, ∠𝐹𝐹 is supplementary to ∠𝐶𝐶, and ∠𝐶𝐶 is supplementary to ∠𝑊𝑊

Given

∠𝑊𝑊 is supplementary to ∠𝐷𝐷, ∠𝐷𝐷 is supplementary to ∠𝐹𝐹, ∠𝐹𝐹 is supplementary to ∠𝐶𝐶, and ∠𝐶𝐶 is supplementary to ∠𝑊𝑊

12

Unit 8: Proving Relationships and Theorems

Unit 8, Lesson 2: Using Properties of Parallelograms Benchmarks MA.912.GR.1.4 Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships, and theorems of parallelograms.

MA.912.LT.4.10 Judge the validity of arguments and give counterexamples to disprove statements.

Learning Target 

I can determine if a quadrilateral is a parallelogram using properties of a parallelogram.

8.2.1 Warm-Up: Bell Work Given: ∠𝑅𝑅𝑅𝑅𝑅𝑅 ≅ ∠𝑅𝑅𝑅𝑅𝑅𝑅 and ∠𝐾𝐾𝐾𝐾𝐾𝐾 ≅ ∠𝐾𝐾𝐾𝐾𝐾𝐾 Prove: 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 is a parallelogram Statement

Reason

1. ∠𝑅𝑅𝑅𝑅𝑅𝑅 ≅ ∠𝑅𝑅𝑅𝑅𝑅𝑅 and ∠𝐾𝐾𝐾𝐾𝐾𝐾 ≅ ∠𝐾𝐾𝐾𝐾𝐾𝐾

1. Given

3.

3. The sum of the angles in a quadrilateral is 360°

2. 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 = 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 and 𝑚𝑚∠𝐾𝐾𝐾𝐾𝐾𝐾 = 𝑚𝑚∠𝐾𝐾𝐾𝐾𝐾𝐾

2. Definition of Congruence

4. 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 + 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 + 4. 𝑚𝑚∠𝐾𝐾𝐾𝐾𝐾𝐾 + 𝑚𝑚∠𝐾𝐾𝐾𝐾𝐾𝐾 = 360° 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 + 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 + 𝑚𝑚∠𝐾𝐾𝐾𝐾𝐾𝐾 + 𝑚𝑚∠𝐾𝐾𝐾𝐾𝐾𝐾 = 360° 5. 2𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 + 2𝑚𝑚∠𝐾𝐾𝐾𝐾𝐾𝐾 = 360° 5. Addition 2𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 + 2𝑚𝑚∠𝐾𝐾𝐾𝐾𝐾𝐾 = 360° 6.

6.

7. ∠𝑅𝑅𝑅𝑅𝑅𝑅 is supplementary to 7. ∠𝐾𝐾𝐾𝐾𝐾𝐾. ∠𝑅𝑅𝑅𝑅𝑅𝑅 is supplementary to ∠𝐾𝐾𝐾𝐾𝐾𝐾 8.

8. If two lines are intersected by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

9. 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 is a parallelogram 9.

Unit 8: Proving Relationships and Theorems

13

8.2.2 Collaboration: Determining Parallelograms

Work with your partner to determine if the figure given is a parallelogram. Finish the statements that follow.

2.

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 is shown with sides ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 and ̅​̅​̅​̅ 𝑁𝑁𝑁𝑁 each measuring ̅​̅​̅​̅ each measuring 9 cm. ̅​̅​̅​̅​̅ and 𝐶𝐶𝐶𝐶 15 cm and sides 𝑊𝑊𝑊𝑊

1. 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 is shown. In the figure, 𝑚𝑚∠𝐶𝐶 = 113°, 𝑚𝑚∠𝐺𝐺 = 67° and 𝑚𝑚∠𝑇𝑇 = 103°.

a.

b.

14

a.

I can conclude …

b.

I know this because …

I can conclude …

I know this because …

Unit 8: Proving Relationships and Theorems

3.

1

In 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆, 𝑚𝑚∠𝑆𝑆 = 120°, and 𝑚𝑚∠𝐹𝐹 = (20𝑥𝑥 + 47)°. If 𝑥𝑥 = ǡ determine whether the figure is a parallelogram.

a.

I can conclude …

b.

I know this because …

2

4. 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 with diagonals ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 and ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃 intersecting at point 𝑅𝑅 is shown.

Given that 𝑀𝑀𝑀𝑀 = 4𝑥𝑥 + 7, 𝐾𝐾𝐾𝐾 = 3𝑦𝑦 − 2, 𝐹𝐹𝐹𝐹 = 4𝑦𝑦 − 3, and 𝑃𝑃𝑃𝑃 = 7𝑥𝑥 − 1, find the values of 𝑥𝑥 and 𝑦𝑦 that make 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 a parallelogram. 𝑥𝑥 = _________________

𝑦𝑦 = ________________

5. When determining if a figure is a parallelogram, what qualities must it have? List them below.

Unit 8: Proving Relationships and Theorems

15

8.2.3 Guided Practice: Is It a Parallelogram?

1.

2.

Use the figure given to fill in the chart below.

If you knew …

...would 𝑯𝑯𝑯𝑯𝑯𝑯𝑯𝑯 be a parallelogram?

∠𝐷𝐷𝐷𝐷𝐷𝐷 ≅ ∠𝑀𝑀𝑀𝑀𝑀𝑀 and ∠𝐵𝐵𝐵𝐵𝐵𝐵 ≅ ∠𝐻𝐻𝐻𝐻𝐻𝐻

o yes o no

̅​̅​̅​̅​̅ ≅ 𝐷𝐷𝐷𝐷 ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

o yes o no

𝑂𝑂 is the midpoint of ̅​̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 and ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

o yes o no

∠𝐷𝐷𝐷𝐷𝐷𝐷 + ∠𝐵𝐵𝐵𝐵𝐵𝐵 = 180° 16

o yes o no

Rocio recently inherited a large field in the shape of a quadrilateral from her Uncle Asher. 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 with diagonals ̅​̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 and ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 intersecting at point 𝐸𝐸 is shown.

She wants to partition the field into four triangular sections to make gardens. While she was out measuring for her supply order, she noticed that the field was shaped like a parallelogram. Her Uncle Asher disagrees. Rocio created the diagram below with the following measurements: 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 23°, 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 72°, 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 49°, and 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 = 23°.

Why or why not?

a.

Who is correct?

b.

How do you know?

Unit 8: Proving Relationships and Theorems

8.2.4 Try It: Parallelograms and Algebra

1.

Quadrilateral 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 is shown where 𝐺𝐺𝐺𝐺 = 3𝑥𝑥 − 1, 𝑆𝑆𝑆𝑆 = 2𝑦𝑦 − 3, 𝐴𝐴𝐴𝐴 = 11, and 𝐺𝐺𝐺𝐺 = 8.

a.

b.

If 𝑥𝑥 = 4 and 𝑦𝑦 = 6, determine whether 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 is a parallelogram.

If 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 is not a parallelogram, what value(s) for 𝑥𝑥 and 𝑦𝑦 would make it a parallelogram?

2.

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 is shown, where diagonals ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 and ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 intersect, 1 such that 𝑊𝑊𝑊𝑊 = −𝑦𝑦 + 4, 𝑅𝑅𝑅𝑅 = 3 𝑥𝑥 + 5, 𝐻𝐻𝐻𝐻 = 5𝑥𝑥 − 6, and 𝐵𝐵𝐵𝐵 = 11 + 𝑦𝑦.

a.

If 𝑥𝑥 = 3 and 𝑦𝑦 = −2, determine whether 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 is a parallelogram.

b.

If 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 is not a parallelogram, what value(s) for 𝑥𝑥 and 𝑦𝑦 would make it a parallelogram?

Unit 8: Proving Relationships and Theorems

17

3. Tariq and Agnes were having a discussion about the different properties of parallelograms. Together they drew and labeled 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇, where 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 = (5𝑥𝑥 − 2)°, 𝑚𝑚∠𝑇𝑇𝑇𝑇𝑇𝑇 = 𝑦𝑦°, and 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 = 118°. What are the values of 𝑥𝑥 and 𝑦𝑦 for Tariq and Agnes’ figure to be a parallelogram?

8.2.5 Wrap-Up: Cool Down

1. Check the property that is always true about the polygon.

Types of Polygons Properties of Figure Quadrilateral Parallelogram Only one set of parallel   sides Two sets of parallel sides





Two sides of equal length





Consecutive angles are supplementary.





Diagonals are congruent.









Diagonals bisect each other.





Opposite angles are congruent.





The sum of all angles is 360°.

18

Unit 8: Proving Relationships and Theorems

Unit 8, Lesson 3: Special Parallelograms

8.3.1 Warm-Up: Estimation 1.

A picture of a bowl with a cereal square is shown.

Benchmarks MA.912.GR.1.4 Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships, and theorems of parallelograms. a.

MA.912.LT.4.10 Judge the validity of arguments and give counterexamples to disprove statements.

2.

Learning Targets  

Estimate the number of cereal squares that will go around the inner circumference of the bowl.

A picture of the bowl with a diameter of cereal squares is shown.

I can prove rectangles are parallelograms.

I can use properties of parallelograms to identify special parallelograms.

a.

Use the diameter of cereal squares to find the circumference.

b.

Compare your estimate to the circumference found using the diameter of cereal squares.

Unit 8: Proving Relationships and Theorems

19

c.

8.3.2 Collaboration: What Is a Rectangle?

1.

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is a rectangle shown on the coordinate grid.

With your partner, list two truths and one lie about this type of quadrilateral in the table below. Share your list with another pair. See if they can spot the lie.

1.

2.

3.

2.

a.

𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 is shown.

List everything you know about this type of quadrilateral.

a. b. b.

20

Using a ruler, draw ̅​̅​̅​̅ 𝑍𝑍𝑍𝑍 and ̅​̅​̅​̅​̅ 𝐾𝐾𝐾𝐾.

These segments are called ______________.

Compare your list with your partner’s list. Add anything you may have missed.

Unit 8: Proving Relationships and Theorems

c.

Measure the length of these segments and write them in the table below. Include units.

Segment

8.3.3 Exploration: Comparing Parallelograms and Rectangles 1.

Length

̅​̅​̅​̅ 𝑍𝑍𝑍𝑍

̅​̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 d.

Make a conjecture about the lengths of the diagonals of a rectangle.

e.

Label the intersection point of the diagonals 𝑆𝑆.

f.

Measure the following segments and fill in their length in the table below. Include units.

Segment

Length

Work with your partner to fill in the table. If the statement describes a property always true of the shape, mark the box.

Statement Parallelogram Rectangle opposite sides   congruent opposite sides   parallel opposite angles   congruent consecutive angles   supplementary diagonals bisect   each other diagonals are   congruent all angles are   congruent Based on the chart, discuss with your partner whether a rectangle could be classified as a parallelogram.

̅​̅ ̅​̅ 𝑍𝑍𝑍𝑍

̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾

̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

g.

What do you notice?

h.

Make a conjecture about what happens to the diagonals of a rectangle when they intersect.

A rectangle is a quadrilateral that contains four right angles. Rectangles may be oblong or square. It’s a member of the following shape classes: polygons, quadrilaterals, trapezoids, parallelograms. Squares form a sub-class.

Unit 8: Proving Relationships and Theorems

21

2.

8.3.4 Guided Instruction: More Special Parallelograms

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 is a rectangle.

Complete the statements.

It is given that 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 is a rectangle. By the definition of a rectangle ∠𝑅𝑅, ∠𝑃𝑃, ∠𝑄𝑄, and ∠𝐷𝐷 are all

o right angles, o acute angles, o obtuse angles,

which means that 𝑚𝑚∠𝑅𝑅 = 𝑚𝑚∠𝑃𝑃 = 𝑚𝑚∠𝑄𝑄 = 𝑚𝑚∠𝐷𝐷 = Since 𝑚𝑚∠𝑃𝑃 + 𝑚𝑚∠𝑄𝑄 = 180° we can show that

o o

and since 𝑚𝑚∠𝐷𝐷 + 𝑚𝑚∠𝑄𝑄 = 180° we can show that by the

o parallelogram o kite

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 is a square with ̅​̅​̅​̅ diagonals ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 and 𝑋𝑋𝑋𝑋 intersecting at point 𝐴𝐴. a.

o 90°Ǥ o 180°.

̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 ∥ ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃 ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 ∥ ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷

̅​̅​̅​̅ o ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 ∥ 𝑃𝑃𝑃𝑃 ̅​̅​̅​̅ ̅​̅​̅​̅ o 𝑅𝑅𝑅𝑅 ∥ 𝐷𝐷𝐷𝐷

o alternate interior angles theorem. o consecutive interior angles theorem.

Therefore, 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 is a

1.

by definition.

A square is a quadrilateral with four right angles and four equal-length sides. It’s a member of the following shape classes: polygons, quadrilaterals, trapezoids, parallelograms, rectangles, rhombi.

Use a ruler and a protractor to fill in the chart with the appropriate measurements.

Include the units.

Segment

Measure

̅​̅​̅​̅ 𝑇𝑇𝑇𝑇

̅​̅​̅​̅ 𝑋𝑋𝑋𝑋

̅​̅​̅​̅ 𝑇𝑇𝑇𝑇

∠𝑋𝑋𝑋𝑋𝑋𝑋

̅​̅​̅​̅ 𝑋𝑋𝑋𝑋

Measure

̅​̅​̅​̅ 𝑅𝑅𝑅𝑅

̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

b.

∠𝑁𝑁𝑁𝑁𝑁𝑁

Which of the following describe the diagonals of a square? Circle all that apply.

congruent

22

Segment/Angle

Unit 8: Proving Relationships and Theorems

bisect

perpendicular

c.

List the features that determine if a parallelogram is a square.

b.

Which of the following describe the diagonals of a rhombus? Select all that apply.

congruent

perpendicular

c.

List the features that determine if a parallelogram is a rhombus.

d.

What are the differences between a square and a rhombus?

A rhombus is a quadrilateral containing four equal-length sides. It’s a member of the following shape classes: polygons, quadrilaterals, trapezoids, parallelograms.

Squares form a sub-class.

2.

bisect

𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 is a rhombus with diagonals ̅​̅​̅​̅​̅ 𝑌𝑌𝑌𝑌 and ̅​̅​̅​̅ 𝑍𝑍𝑍𝑍 intersecting at point 𝐻𝐻. a.

Use a ruler and a protractor to fill in the chart with the appropriate measurements.

Include units.

Segment ̅​̅​̅​̅​̅ 𝑌𝑌𝑌𝑌 ̅​̅​̅​̅ 𝑍𝑍𝑍𝑍

̅​̅​̅​̅ 𝑌𝑌𝑌𝑌

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊

Measure

Segment/Angle

Measure

̅​̅​̅​̅ 𝑍𝑍𝑍𝑍

̅​̅​̅​̅ 𝐷𝐷𝐷𝐷

∠𝑍𝑍𝑍𝑍𝑍𝑍 ∠𝑌𝑌𝑌𝑌𝑌𝑌

Unit 8: Proving Relationships and Theorems

23

3.

Use the numbered statements in the box to fill in the Venn diagram comparing rectangles, squares, and rhombi.

1. opposite sides are congruent

6. consecutive angles are supplementary

2. opposite sides are parallel

7. all sides are congruent

3. all angles are congruent

8. diagonals are congruent

4. diagonals are perpendicular

9. diagonals bisect each other

5. opposite angles are congruent

Rectangle

8.3.5 Bring It Together: Which Special Parallelogram Is It? 1.

Explain how to show a parallelogram is a rectangle but not a rhombus or a square.

2.

Explain how to show a parallelogram is a rhombus but not a square or a rectangle.

Rhombus

3. Explain how to show a parallelogram is a square.

Square

24

Unit 8: Proving Relationships and Theorems

Unit 8, Lesson 4: Kites and Parallelograms

8.3.6 Wrap-Up: Reflection 1.

Plot a point on the grid to indicate your level of interest and how much sense you made of today’s learning targets.

Benchmark MA.912.GR.1.4 Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships, and theorems of parallelograms.

Learning Target 

2.

I can use properties of parallelograms, kites, and special parallelograms to solve mathematical and real-world problems.

Explain why you placed the point where you did.

Unit 8: Proving Relationships and Theorems

25

8.4.1 Warm-Up: Canstruction

8.4.2 Exploration: What Is a Kite?

Each year just before Thanksgiving there are competitions held all over the world to build unique Canstructions with canned food that is collected from the community.

The food is then given to local food banks. The picture shows the Canstruction called "Downside Up," which was the winning exhibit from Canstruction 2010 in New York City.

1.

What do you wonder about “Downside Up”?

2.

Make an educated guess at the number of cans used to build “Downside Up”. Show your work.

3.

Compare your educated guess with your partner’s.

Describe how close your guess was to your partner’s guess.

26

̅​̅ and ̅​̅​̅​̅​̅ Kite 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 with diagonals ̅​̅ 𝑆𝑆𝑆𝑆 𝑊𝑊𝑊𝑊 intersecting at point 𝑉𝑉 ̅​̅​̅​̅ and ̅​̅​̅​̅ and 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 with diagonals 𝐹𝐹𝐹𝐹 𝐻𝐻𝐻𝐻 intersecting at point 𝐾𝐾 are shown.

Unit 8: Proving Relationships and Theorems

1.

Using a ruler, work with a partner to fill in the chart below with the appropriate measurements. Each partner should find the measurements for one kite and share their results with the other.

Segment ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆

̅​̅​̅​̅​̅ 𝑆𝑆𝑆𝑆

𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺

Measurement

Segment

̅​̅ ̅​̅ 𝑆𝑆𝑆𝑆

̅​̅​̅​̅ 𝐹𝐹𝐹𝐹

̅​̅​̅​̅ 𝑇𝑇𝑇𝑇

̅​̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊

̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

̅​̅​̅​̅ 𝑆𝑆𝑆𝑆

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅

These segment pairs are adjacent sides of a kite. a.

Make a conjecture about the measures of adjacent sides of a kite.

b.

Make a conjecture about the lengths of diagonals of a kite.

Measurement 3.

̅​̅​̅​̅ 𝐹𝐹𝐹𝐹

̅​̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊

𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭

̅​̅​̅​̅ 𝐹𝐹𝐹𝐹

̅​̅​̅​̅ 𝑅𝑅𝑅𝑅

2.

̅​̅​̅​̅ 𝐿𝐿𝐿𝐿

Using a protractor, work with your partner to fill in the chart below with the appropriate measurements. Each partner should find the measurements for one kite and share their results with the other.

Angle

̅​̅​̅​̅ 𝐹𝐹𝐹𝐹

𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺

Measurement

∠𝑆𝑆𝑆𝑆𝑆𝑆

∠𝑊𝑊𝑊𝑊𝑊𝑊

̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

∠𝑅𝑅𝑅𝑅𝑅𝑅 ∠𝑇𝑇𝑇𝑇𝑇𝑇

̅​̅​̅​̅ 𝐿𝐿𝐿𝐿

a.

What do you notice about ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 and ̅​̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 in 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 and ̅​̅​̅​̅ and 𝐹𝐹𝐹𝐹 ̅​̅​̅​̅ in 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹? 𝐹𝐹𝐹𝐹

b.

̅​̅​̅​̅ and ̅​̅​̅​̅​̅ What do you notice about 𝑅𝑅𝑅𝑅 𝑊𝑊𝑊𝑊 in 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 and ̅​̅​̅​̅​̅ ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 and 𝐿𝐿𝐿𝐿 in 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹?

∠𝑇𝑇𝑇𝑇𝑇𝑇

Angle

𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭

Measurement

∠𝐹𝐹𝐹𝐹𝐹𝐹

∠𝐻𝐻𝐻𝐻𝐻𝐻

∠𝐿𝐿𝐿𝐿𝐿𝐿

∠𝐹𝐹𝐹𝐹𝐹𝐹 ∠𝐹𝐹𝐹𝐹𝐹𝐹

a.

Make a conjecture about what happens when the diagonals of a kite intersect.

b.

Make a conjecture about the measures of opposite angles of a kite.

Unit 8: Proving Relationships and Theorems

27

8.4.3 Collaboration: Quadrilaterals

A kite is a quadrilateral with exactly two pairs of adjacent congruent sides. ͳǤ A kite has the following properties: o One pair of opposite congruent angles o Perpendicular diagonals o One bisected diagonal

With your partner, use the bank of phrases to fill in the chart comparing quadrilaterals. Statements can be used more than once.

A. Opposite sides are congruent.

C. All sides are congruent.

E. Opposite angles are congruent.

G. All angles are congruent.

I. Two pairs of adjacent sides are congruent.

K. One pair of opposite angles are congruent.

M. One pair of parallel sides O. Two pairs of congruent angles a.

Write the letter(s) from the bank.

Quadrilateral Parallelogram Rectangle Square Rhombus Kite

28

B. Opposite sides are parallel.

D. Diagonals bisect each other.

F. Diagonals are congruent.

H. Diagonals are perpendicular.

J. One diagonal is bisected.

L. Opposite angles are supplementary.

N. Non-parallel sides are congruent.

Unit 8: Proving Relationships and Theorems

Sides

Angles

Diagonals

b.

Compare your chart with that of another pair. Add in anything you may have missed.

8.4.4 Guided Instruction: Quadrilaterals and Algebra 1.

c.

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 with diagonals ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵 and ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 intersecting at point 𝐴𝐴 is shown. Find the value of 𝑥𝑥 that makes 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 a kite if 𝐵𝐵𝐵𝐵 = 3𝑥𝑥 − 1, 𝐿𝐿𝐿𝐿 = 17, 𝑀𝑀𝑀𝑀 = 2𝑥𝑥, and 𝑈𝑈𝑈𝑈 = 23.

Discuss with your partner whether or not kites can be classified as parallelograms.

Unit 8: Proving Relationships and Theorems

29

2.

Find the values of 𝑥𝑥 and 𝑦𝑦 that make 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 a kite if 𝑃𝑃𝑃𝑃 = 5𝑥𝑥 − 4, 𝑃𝑃𝑃𝑃 = 8𝑦𝑦 − 4, 𝑆𝑆𝑆𝑆 = 3𝑦𝑦 + 1, and 𝑅𝑅𝑅𝑅 = 36.

3.

The Dockland office building is one of the most famous buildings in Hamburg, Germany. Built in 2005, this building houses roughly 9,000 square meters of office space and breathtaking views of the Elbe River, but the size and the views are not what makes this building so famous. The building itself is in the shape of a parallelogram, 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻.

The perimeter of the Dockland building measures approximately 238 meters. If the length of ̅​̅​̅​̅ 𝐹𝐹𝐹𝐹 is 2.8 times the length of ̅​̅​̅​̅ 𝐹𝐹𝐹𝐹, what are the measures of ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻, ̅​̅​̅​̅ , 𝐹𝐹𝐹𝐹 ̅​̅​̅​̅ , and 𝐹𝐹𝐹𝐹 ̅​̅​̅​̅? Round your answers to the nearest 𝑆𝑆𝑆𝑆 thousandth of a meter.

30

Unit 8: Proving Relationships and Theorems

2.

8.4.5 Your Turn!

1.

Oliver’s carpentry company, Wood 4 You, has been hired to build a fence around a vegetable garden. The gate must be a perfect rectangle. The gate Oliver built is shown, where quadrilateral 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 outlines the gate.

Piper is entering a kite-making contest. The kite she designed is shown, where the kite has been labeled as 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 with measurements 𝐸𝐸𝐸𝐸 = 7.5, 𝑌𝑌𝑌𝑌 = 6, and 𝐹𝐹𝐹𝐹 = 10.2, given in inches.

Piper wishes to add a strip of glitter fabric along diagonals ̅​̅​̅​̅ 𝐸𝐸𝐸𝐸 and ̅​̅​̅​̅​̅ 𝑌𝑌𝑌𝑌 that will overlap at their intersection point 𝑁𝑁, as well as along the perimeter of the kite. How many inches of glitter fabric are needed to cover the diagonals and the perimeter of the kite?

Oliver wants to be absolutely sure that the gate is a rectangle. Describe two different ways Oliver can use his tape measure to ensure the gate is a true rectangle.

Unit 8: Proving Relationships and Theorems

31

Unit 8, Lesson 5: Trapezoids

8.4.6 Wrap-Up: Cool Down 1.

Kite 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 is shown. Find the value of 𝑥𝑥 if 𝑚𝑚∠𝑁𝑁 = 53°, 1 𝑚𝑚∠𝑇𝑇 = 67°, and 𝑚𝑚∠𝐹𝐹 = ( 𝑥𝑥 − 8) °.

Benchmarks

2

MA.912.GR.1.5 Prove relationships and theorems about trapezoids. Solve mathematical and real-world problems involving postulates, relationships, and theorems of trapezoids.

MA.912.LT.4.10 Judge the validity of arguments and give counterexamples to disprove statements.

Learning Targets  

32

I can prove relationships and theorems about trapezoids.

I can use relationships and theorems about trapezoids.

Unit 8: Proving Relationships and Theorems

8.5.2 Collaboration: Properties of Isosceles Trapezoids

8.5.1 Warm-Up: Trapezoid or Trapezium?

1.

Which quadrilateral is classified as a trapezoid?

1.

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 and 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 are both isosceles trapezoids. ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 ̅​̅​̅​̅​̅ and 𝑊𝑊𝑊𝑊 are the diagonals of 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 and intersect at ̅​̅​̅​̅ are the diagonals of 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 and point 𝑌𝑌. ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 and 𝐾𝐾𝐾𝐾 intersect at point 𝑁𝑁.

There is more than one definition of a trapezoid.

Inclusive Definition Exclusive Definition A quadrilateral with at least A quadrilateral with exactly one pair of parallel opposite one pair of parallel opposite sides. sides.

There is more than one definition of trapezium.

American Definition A quadrilateral with no parallel sides.

2.

British Definition A quadrilateral with two sides parallel.

Use the definitions to classify quadrilaterals 1, 2, and 3.

Write the numbers of the quadrilateral in the table.

Inclusive Definition of Trapezoid

Exclusive Definition of Trapezoid

American Definition of Trapezium

British Definition of Trapezium

a.

Using a ruler and a protractor to measure 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 and 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆, work with your partner to determine if the following conjectures about isosceles trapezoids are true or false. If false, write a new conjecture based on your measurements.

Unit 8: Proving Relationships and Theorems

33

Conjecture All angles of an isosceles trapezoid are congruent

True or False?

If false, write a new conjecture.

2.

∠𝑅𝑅𝑅𝑅𝑅𝑅 and ∠𝑊𝑊𝑊𝑊𝑊𝑊 in 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 are called base angles. ∠𝑆𝑆𝑆𝑆𝑆𝑆 and ∠𝐾𝐾𝐾𝐾𝐾𝐾 are base angles in 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆.

Based on your previous measurements, make a conjecture about the base angles of an isosceles trapezoid.

Opposite sides are parallel

3. Each isosceles trapezoid has two pairs of base angles.

Opposites angles are congruent

a.

What is the other pair of base angles in trapezoid 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅?

b.

What is the other pair of base angles in trapezoid 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆?

Opposite sides are congruent

Diagonals bisect each other

b.

34

Compare your chart to that of another pair. Discuss any differences. Make corrections as necessary.

Unit 8: Proving Relationships and Theorems

A trapezoid is a quadrilateral with at least one pair of parallel sides. It’s a member of the following shape classes: polygons, quadrilaterals. Its sub-classes include parallelograms, rectangles, rhombuses, and squares.

8.5.3 Exploration: Parts of a Trapezoid 1.

Trapezoid 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 is shown.

An isosceles trapezoid is a trapezoid with at least one pair of congruent base angles.

a.

b.

The parallel sides of a trapezoid are called the bases. The trapezoid shown has one pair of bases.

Label each base red and mark them to show they are parallel on 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺.

The non-parallel sides of a trapezoid are called the legs. The trapezoid shown has two legs.

Label each leg of 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺.

The median of a trapezoid connects the nonparallel sides at their midpoints. The median is sometimes referred to as the midsegment.

c.

Using your ruler, plot the midpoint of ̅​̅​̅​̅​̅ 𝐺𝐺𝐺𝐺 and the midpoint of ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊. Connect these points to create the median of trapezoid 𝐺𝐺𝑊𝑊𝑆𝑆𝑆𝑆 in green. Be sure to mark the legs appropriately showing that the median is connected at the midpoints.

Unit 8: Proving Relationships and Theorems

35

An altitude of a trapezoid is any segment from a point on one of the bases perpendicular to the line containing the other base.

d.

8.5.4 Guided Instruction: Proving Isosceles Trapezoid Properties 1.

The length of the altitude is the distance between the bases.

̅​̅​̅​̅​̅. ̅​̅​̅​̅​̅ and 𝐶𝐶𝐶𝐶 Trapezoid 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 is shown with altitudes 𝐾𝐾𝐾𝐾

Draw the altitude on 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺. Name the altitude by naming the endpoints and mark the trapezoid to show that the altitude is perpendicular to the base.

2.

Trapezoid 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is shown with midsegment ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 and ̅​̅​̅​̅ altitude 𝑇𝑇𝑇𝑇.

a.

Trace the trapezoid onto a piece of patty paper.

b.

Fold the patty paper so that vertices 𝐾𝐾 and 𝐶𝐶 coincide.

c.

Describe what else you notice when you fold the patty paper.

d.

Work with a partner to list all of the congruent parts of trapezoid 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾.

e.

Compare your list with that of another pair. Add any congruent parts you may have missed.

f.

Describe how the reflecting across the midpoint of ̅​̅​̅​̅ shows that 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 can be classified as an 𝐾𝐾𝐾𝐾 isosceles trapezoid.

Select all of the information that is true about 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴.

 ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 ≅ ̅​̅​̅​̅ 𝐴𝐴𝐴𝐴

 𝑚𝑚∠𝐿𝐿𝐿𝐿𝐿𝐿 = 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹

̅​̅​̅​̅  ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 ⊥ 𝐴𝐴𝐴𝐴

̅​̅​̅​̅  ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 ≅ 𝑆𝑆𝑆𝑆

 ̅​̅​̅​̅ 𝐴𝐴𝐴𝐴 ∥ ̅​̅​̅​̅ 𝐹𝐹𝐹𝐹

 ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 ∥ ̅​̅​̅​̅ 𝐹𝐹𝐹𝐹

 ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 ⊥ ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊

 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷  𝑚𝑚∠𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷

36

Unit 8: Proving Relationships and Theorems

2.

Given: 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 is an isosceles

3.

trapezoid with ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 ∥ ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃

Given: 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 is an isosceles trapezoid with ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵 ∥ ̅​̅​̅​̅​̅ 𝑋𝑋𝑋𝑋

Prove: ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 ≅ ̅​̅​̅​̅​̅ 𝐾𝐾𝐾𝐾

Prove: ∠𝑅𝑅𝑅𝑅𝑅𝑅 and ∠𝑅𝑅𝑅𝑅𝑅𝑅

are supplementary, ∠𝐵𝐵𝐵𝐵𝐵𝐵

and ∠𝐵𝐵𝐵𝐵𝐵𝐵 are supplementary Complete the following two-column proof.

Statement 1.

Reason 1.

Given that 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 is an isosceles trapezoid, then ________________ because base angles of an

isosceles trapezoid are congruent. Since it is given that 2. ̅​̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 ≅ ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾

2.

3.

3. Base angles of an isosceles trapezoid are congruent

_________, then ∠𝑅𝑅𝑅𝑅𝑅𝑅 and ∠𝐵𝐵𝐵𝐵𝐵𝐵 are

________________ and ∠______________ and ∠______________

are supplementary because when parallel lines are cut by a transversal, _____________________________

4. ̅​̅​̅​̅ 𝑁𝑁𝑁𝑁 ≅ ̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

4.

5. ∆𝐻𝐻𝐻𝐻𝐻𝐻 ≅ ∆𝐾𝐾𝐾𝐾𝐾𝐾

5. SAS Congruence Theorem

6.

6. CPCTC

are supplementary. Therefore, by definition of supplementary angles, 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 + 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 =_____ and ___________________= 180°. By substitution,

𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 + 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 = 180° and ___________________.

Therefore, by ________________________________, ∠𝑅𝑅𝑅𝑅𝑅𝑅 and ∠𝑅𝑅𝑅𝑅𝑅𝑅 are supplementary and ∠𝐵𝐵𝐵𝐵𝐵𝐵 and ∠𝐵𝐵𝐵𝐵𝐵𝐵 are supplementary.

Unit 8: Proving Relationships and Theorems

37

8.5.5 Guided Practice: Isosceles Trapezoids 1.

2.

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 is an isosceles trapezoid.

8.5.6 Wrap-Up: Ticket Out the Door 1.

a.

̅​̅​̅​̅ = 3𝑥𝑥 − 7, find If diagonal ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 = 18 and diagonal 𝑅𝑅𝑅𝑅 the value of 𝑥𝑥.

b.

Diagonals ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 and ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 intersect at point 𝐾𝐾. If 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 = 20.6°, 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 = 24.4°, and 𝑚𝑚∠𝐵𝐵𝐵𝐵𝐵𝐵 = 138.9°, what is 𝑚𝑚∠𝑊𝑊𝑊𝑊𝑊𝑊?

̅​̅​̅​̅ and Isosceles trapezoid 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 is shown with altitudes 𝐹𝐹𝐹𝐹 ̅​̅​̅​̅ 𝑉𝑉𝑉𝑉 .

If 𝐹𝐹𝐹𝐹 = 12, 𝑇𝑇𝑇𝑇 = 36√5, and 𝑉𝑉𝑉𝑉 = 8, what is the length of ̅​̅​̅​̅ 𝐹𝐹𝐹𝐹?

Isosceles trapezoid 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 has altitudes ̅​̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 and ̅​̅​̅​̅​̅ 𝐶𝐶𝐶𝐶. If 𝐾𝐾𝐾𝐾 = 6, 𝐴𝐴𝐴𝐴 = 30, and 𝐾𝐾𝐾𝐾 = 15, what is the length of altitude ̅​̅​̅​̅​̅ 𝐶𝐶𝐶𝐶?

38

Unit 8: Proving Relationships and Theorems

Unit 8, Lesson 6: Solving Real-World Problems With Quadrilaterals Benchmarks MA.912.GR.1.4 Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships, and theorems of parallelograms.

8.6.1 Warm-Up: Error Analysis Kite 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 is shown, with 𝑚𝑚∠𝑊𝑊 = 90°, 𝑚𝑚∠𝐶𝐶 = 68°, and 𝑚𝑚∠𝑁𝑁 = 6𝑥𝑥 + 5.

Marcia and Eugene both worked to determine the value of 𝑥𝑥. Their work is shown.

MA.912.GR.1.5 Prove relationships and theorems about trapezoids. Solve mathematical and real-world problems involving postulates, relationships, and theorems of trapezoids.

Learning Targets  

I can use properties of parallelograms to solve realworld problems.

I can use properties of trapezoids to solve real-world problems.

1.

Marcia 90 + 90 + 68 + 6𝑥𝑥 + 5 = 360 6𝑥𝑥 + 5 + 248 = 360 6𝑥𝑥 + 253 = 360 6𝑥𝑥 = 107 107 𝑥𝑥 = 6

Eugene 360 − (90 + 68) = 202 6𝑥𝑥 + 5 = 202 6𝑥𝑥 = 197 197 𝑥𝑥 = 6

Marcia

Eugene

Evaluate each student’s work, then write a note to each student explaining the error they made.

Unit 8: Proving Relationships and Theorems

39

8.6.2 Activity: Word Problem Gallery Walk 1.

Work through the word problems with your partner.

Show your work and write your answers in the appropriate boxes below.

4

Problem Number

Work

1

Answer

𝑦𝑦 =

𝐾𝐾𝐾𝐾 = 𝐹𝐹𝐹𝐹 = 5 Area =

2

𝑅𝑅𝑅𝑅 = 𝑇𝑇𝑇𝑇 =

𝑚𝑚∠𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 =

3

6

𝑚𝑚∠𝐺𝐺𝐺𝐺𝐺𝐺 = 𝑚𝑚∠𝑀𝑀𝑀𝑀𝑀𝑀 =

40

Unit 8: Proving Relationships and Theorems

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 ________ a parallelogram.

8.6.3 Your Turn!

7

1.

I am a quadrilateral with two pairs of opposite parallel sides. At least one of my angles has a measure of 90° and both of my diagonals measure 12 inches. Select all shapes that match this description.  Parallelogram  Rectangle  Square  Rhombus  Kite

2.

In your own words, write a description of a parallelogram that could only match a square.

3.

If a quadrilateral is a parallelogram, what additional information would be needed to show that the shape is a rectangle?

4.

If a quadrilateral is a parallelogram, what additional information would be needed to show that the shape is a rhombus?

The dimensions of 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 are

8 𝐷𝐷𝐷𝐷 =

Unit 8: Proving Relationships and Theorems

41

Unit 8, Lesson 7: Trapezoid Midsegment Theorem

8.6.4 Wrap-Up: Reflection 1.

Circle the icon that best describes your understanding of each concept below.

I need help.

I got it.

I could help someone.

Benchmark MA.912.GR.1.5 Prove relationships and theorems about trapezoids. Solve mathematical and real-world problems involving postulates, relationships, and theorems of trapezoids.

Properties of a parallelogram Learning Targets

Properties of a rectangle Properties of a kite

 

Completing proofs using properties and theorems about parallelograms Using the properties of a parallelogram to solve problems Properties of a trapezoid Using the properties of a trapezoid to solve problems

42

I can prove the trapezoid midsegment theorem.

I can use properties of trapezoids to solve mathematical and real-world problems.

Unit 8: Proving Relationships and Theorems

8.7.1 Warm-Up: Trapezoids in Architecture

8.7.2 Exploration: Midsegment of a Trapezoid 1.

Four different buildings are shown.

Machu Picchu, Peru

Greenland Center, Beijing, China

Garden Terrace, Nagasaki

International Spy Museum, Washington D.C.

̅​̅​̅​̅ ∥ ̅​̅​̅​̅ Trapezoid 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 is shown, where 𝑅𝑅𝑅𝑅 𝑌𝑌𝑌𝑌 .

a.

Use patty paper or a ruler to determine if ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 is the midsegment of trapezoid 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅Ǥ Justify your conclusion.

b.

Use a ruler to complete the table.

Segment(s)

1.

2.

What do you notice about the buildings?

Describe how a trapezoid is used in each building.

Measure

̅​̅​̅​̅ 𝑅𝑅𝑅𝑅

̅​̅​̅​̅ 𝑌𝑌𝑌𝑌 ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇

Unit 8: Proving Relationships and Theorems

43

2.

What do you notice about the relationship between the length of the midsegment and the sum of the lengths of the bases?

8.7.3 Guided Instruction: Trapezoid Midsegment Theorem

1.

3.

Write an equation to show the relationship between the length of the midsegment of a trapezoid and the length of its bases.

The midsegment of a triangle is a segment joining the midpoints of two sides of a triangle. ∆𝐴𝐴𝐴𝐴𝐴𝐴 is shown with midsegment ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 .

Fill in the blanks to complete the triangle midsegment theorem.

The triangle midsegment theorem states that the midsegment of a triangle is ____________ to the third side it does not touch and is ________ the length of the third side.

44

Unit 8: Proving Relationships and Theorems

2.

∆𝐹𝐹𝐹𝐹𝐹𝐹 with midsegment ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 and trapezoid 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 are ̅​̅​̅​̅ ∥ 𝐾𝐾𝐾𝐾 ̅​̅​̅​̅​̅. shown where 𝐹𝐹𝐹𝐹

a.

Explain why 𝑆𝑆𝑆𝑆 = 𝐾𝐾𝐾𝐾.

b.

If ̅​̅​̅​̅ 𝑁𝑁𝑁𝑁 ≅ ̅​̅​̅​̅ 𝐹𝐹𝐹𝐹, justify why ̅​̅​̅​̅ 𝑁𝑁𝑁𝑁 ≅ ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅.

d.

Discuss with your partner what else needs to be ̅​̅​̅​̅ is a determined so that it can be stated that 𝑆𝑆𝑆𝑆 midsegment of 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹. Summarize your discussion.

Trapezoid Midsegment Theorem

c.

Discuss with your partner what information is necessary so that the definition of a trapezoid can be used to state that 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 is a trapezoid.

Summarize your discussion.

The midsegment of a trapezoid is parallel to the bases and is half the sum of the bases.

Unit 8: Proving Relationships and Theorems

45

8.7.4 Guided Practice: Using the Trapezoid Midsegment Theorem 1.

If 𝐹𝐹𝐹𝐹 = 11.4 and 𝑌𝑌𝑌𝑌 = 17.6, what is the length of ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 ?

d.

What is the value of 𝑘𝑘 if 𝑊𝑊𝑊𝑊 = 4𝑘𝑘 + 4.2, 𝐹𝐹𝐹𝐹 = 7𝑘𝑘 + 2.1, and 𝑌𝑌𝑌𝑌 = 6𝑘𝑘 + 16.8?

e.

If 𝑊𝑊𝑊𝑊 = 3 𝑝𝑝 and 𝑊𝑊𝑊𝑊 = 12 − 2𝑝𝑝, what is the length of ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊?

̅​̅​̅​̅ is shown.

Trapezoid 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 with midsegment 𝐹𝐹𝐹𝐹

a.

b.

46

c.

̅​̅​̅​̅?

If 𝑊𝑊𝑊𝑊 = 14.2 and 𝑌𝑌𝑌𝑌 = 29.7, what is the length of 𝐹𝐹𝐹𝐹

If 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 is an isosceles trapezoid and 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = 53°, what is 𝑚𝑚∠𝑊𝑊𝑊𝑊𝑊𝑊?

2

Unit 8: Proving Relationships and Theorems

2.

8.7.5 Your Turn!

1.

̅​̅​̅​̅ . Find Trapezoid 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 is shown with midsegment 𝑈𝑈𝑈𝑈 the values of 𝑥𝑥 and 𝑦𝑦 if 𝑅𝑅𝑅𝑅 = −2𝑥𝑥 − 2𝑦𝑦, 𝐴𝐴𝐴𝐴 = 10, 𝑅𝑅𝑅𝑅 = 4𝑥𝑥, 𝑈𝑈𝑈𝑈 = 14, and 𝐴𝐴𝐴𝐴 = −𝑦𝑦 + 8.

Located on the Nature Coast State Trail in north-central Florida is the Suwanee River Trestle, a bridge that is shaped like an isosceles trapezoid. The state has been inspecting older bridges to ensure their safety and has determined that the Suwanee River Trestle, represented as trapezoid 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍, needs an additional support brace through its middle.

̅​̅​̅​̅​̅ at The support brace is to connect sides ̅​̅​̅​̅ 𝑍𝑍𝑍𝑍 and 𝑌𝑌𝑌𝑌 their midpoints. If the length of leg 𝑍𝑍𝑍𝑍 = 28 feet (ft) and the perimeter of the trestle measures 153 ft, what is the length, in ft, of the new support brace?

Unit 8: Proving Relationships and Theorems

47

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

8.7.6 Wrap-Up: Cool Down 1.

48

Your friend is absent today and will need the notes from class. Write a quick note to help them understand the trapezoid midsegment theorem.

Unit 8: Proving Relationships and Theorems

Unit 9 - Two- and Three-Dimensional Shapes 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

Lesson Finding the Area of Two-Dimensional Figures Density Based on Area Cross Sections Solids of Rotation Cavalieri’s Principle Finding Volume Volume in the Real World Surface Area of Three-Dimensional Figures Surface Area in the Real World Determining How Dilations Affect Area Determining Surface Area Given Dimension Changes

Unit 9: Two- and Three-Dimensional Shapes

Page Number 51 59 65 70 76 82 88 93 100 104 112

49

The following B.E.S.T Standards will be covered in this unit: MA.912.GR.4.1 Identify the shapes of two-dimensional cross-sections of three-dimensional figures.

• Clarification 1: Instruction includes the use of manipulatives and models to visualize cross-sections.

• Clarification 2: Instruction focuses on cross-sections of right cylinders, right prisms, right pyramids and right cones that are parallel or perpendicular to the base.

MA.912.GR.4.2 Identify three-dimensional objects generated by rotations of two-dimensional figures.

• Clarification 1: The axis of rotation must be within the same plane but outside of the given two-dimensional figure.

MA.912.GR.4.3 Extend previous understanding of scale drawings and scale factors to determine how dilations affect the area of two-dimensional figures and the surface area or volume of three-dimensional figures.

MA.912.GR.4.4 Solve mathematical and real-world problems involving the area of two-dimensional figures.

• Clarification 1: Instruction includes concepts of population density based on area.

MA.912.GR.4.5 Solve mathematical and real-world problems involving the volume of three-dimensional figures limited to cylinders, pyramids, prisms, cones and spheres.

• Clarification 1: Instruction includes concepts of density based on volume.

• Clarification 2: Instruction includes using Cavalieri’s principle to give informal arguments about the formulas for the volumes of right and non-right cylinders, pyramids, prisms and cones.

MA.912.GR.4.6 Solve mathematical and real-world problems involving the surface area of three-dimensional figures limited to cylinders, pyramids, prisms, cones and spheres.

50

Unit 9: Two- and Three-Dimensional Shapes

Unit 9, Lesson 1: Finding the Area of Two-Dimensional Figures

Benchmark

9.1.1 Warm-Up: Michael Jordan Basketball legend Michael Jordan is often quoted as saying, “I’ve failed over and over and over again in my life and that is why I succeed.”

1.

What is a skill, activity, or subject that you have failed at?

2.

Explain Jordan’s quote in your own words.

3.

How can you turn a failure into a success story?

MA.912.GR.4.4 Solve mathematical and real-world problems involving the area of two-dimensional figures.

Learning Target 

I can solve mathematical and real-world problems involving the area of two-dimensional figures.

Unit 9: Two- and Three-Dimensional Shapes

51

9.1.2 Collaboration: Finding Area 1.

3.

Parallelogram 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 is shown on the coordinate grid.

Complete the table.

Figure

Area Formula

Rectangle Triangle 2.

How is the area of a triangle related to the area of a rectangle?

Tyrese could not remember how to find the area of a parallelogram so he divided the parallelogram into 2 triangles and a rectangle. Work with your partner to use Tyrese’s method to find the area of 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵.

Shape Triangle 1 Triangle 2 Rectangle Total Area of Parallelogram

52

Unit 9: Two- and Three-Dimensional Shapes

Area

4.

Sarianna rearranged the parallelogram to create a rectangle.

5.

Find the area of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 using the division shown on the grid.

Use Sarianna’s rectangle to find the area of 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵.

Unit 9: Two- and Three-Dimensional Shapes

53

6.

𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 is shown on the coordinate grid.

7.

and 𝑏𝑏2 are the bases. Use the formula Ava found to find the area of 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹.

8.

a.

Ava found a formula for the area of a trapezoid. The 1 formula is 𝐴𝐴 = 2 ℎ(𝑏𝑏1 + 𝑏𝑏2 ), where ℎ is the height and 𝑏𝑏1

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 is shown on the coordinate grid.

Divide the quadrilateral into two triangles and a rectangle. Then, find the area of each.

Shape

Area

Triangle 1 Triangle 2 Rectangle b.

c.

54

What is the area of 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹?

a.

What type of quadrilateral is 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇?

What type of quadrilateral is 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹?

Unit 9: Two- and Three-Dimensional Shapes

b.

Determine the area of 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇. Show your work or explain your thinking.

The area of a rhombus can be found using the formula 1

𝐴𝐴 = 𝑝𝑝𝑝𝑝, where 𝑝𝑝 and 𝑞𝑞 are the lengths of the diagonals.

2

d.

c.

Complete the table.

Line Segment ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇

̅​̅​̅​̅ 𝐾𝐾𝐾𝐾

Length

Use the formula to find the area of 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇.

Formulas for area can easily be forgotten, so it is always helpful to be able to break a figure down into triangles and rectangles in order to find the area. The exception is finding the area of a circle or a part of a circle.

9.

What is the area of a circle?

Unit 9: Two- and Three-Dimensional Shapes

55

10. The drama program at school is building a new semicircular stage. The distance across the stage is 30 feet.

What is the area of the new stage? Round to the nearest thousandth.

56

9.1.3 Guided Practice: Using the Formulas 1.

The radius of a circle is 7 feet (ft). Find the area, in feet, of the circle. Leave the answer in 𝜋𝜋 form.

2.

The base of the parallelogram is 22 inches (in). The area of the parallelogram is 132 inches. Find the height, in inches, of the parallelogram.

Unit 9: Two- and Three-Dimensional Shapes

3.

The diagonals of a rhombus are ͶͲ centimeters (cm) and ͳͺ centimeters. Find the area, in centimeters, of the rhombus.

4.

A composite shape of a rhombus, a parallelogram, and a triangle is shown.

Complete the table. Round to the nearest thousandth.

Shape

Area

Rhombus Parallelogram Triangle Composite Area

Unit 9: Two- and Three-Dimensional Shapes

57

9.1.4 Your Turn!

1.

9.1.5 Wrap-Up: Reflection

The design of a room is shown, where 1 unit = 5 feet.

1.

Circle the icon that best describes your understanding of each.

I need help.

I got it.

I could help someone.

Finding the area of a triangle Finding the area of a rectangle Finding the area of a trapezoid Finding the area of a square Determine the area, in square feet, of the room. Round to the nearest thousandth.

Finding the area of a rhombus Finding the area of a parallelogram Finding the area of a circle Finding the area of a composite figure

58

Unit 9: Two- and Three-Dimensional Shapes

Unit 9, Lesson 2: Density Based on Area Benchmark

MA.912.GR.4.4 Solve mathematical and real-world problems involving the area of two-dimensional figures.

Learning Target

9.2.1 Warm-Up: Would You Rather?

1.

Car A $19,000

12% depreciation per year when you upgrade the car after 5 years.

2. 

Which car would you rather buy?

Explain your answer.

Car B $25,000

16% depreciation per year when you upgrade the car after 5 years.

I can solve real-world problems that involve density using area.

Unit 9: Two- and Three-Dimensional Shapes

59

9.2.2 Collaboration: Population Density 1.

2.

Discuss with your partner your understanding of population density. Describe the population density for the city or town you live in.

3.

Make predictions about the population density for Florida’s five largest cities.

The five largest cities in Florida and their populations are shown.

City Jacksonville Miami Tampa Orlando St. Petersburg

Population 890,467 454,279 387,916 280,832 261,338

Largest Population Density

The table shows the area of Florida’s five largest cities.

City Jacksonville Miami Tampa Orlando St. Petersburg

Area (square miles) 747 35.87 113.41 102.4 61.74

What Do You Notice About the Population and the Area?

4.

Explain your thinking.

5.

Ask a classmate for the predictions they made as well as their reasons. Record their cities and write your summary of their reason. You are the only person who should write in the middle two columns of the table.

Have the person initial next to your summary stating that your summary was correct.

City

Largest Population Density Smallest Population Density 60

Smallest Population Density

Unit 9: Two- and Three-Dimensional Shapes

My Summary of Their Reason

Initials

To find population density, divide the population by the area.

Population Density =

6.

population area

9.2.3 Guided Instruction: Area and Density 1.

An outline of the city of Miramar, Florida, with measurements in miles (mi), is shown.

Determine the population density, in persons per square mile (persons/mi2 ), of the five largest cities in Florida and record them in the table from largest to smallest. Round answers to the nearest whole number.

City

Population Density (persons/mi.2)

a.

What is the area of the city, in square miles?

b.

Determine the population density of Miramar for 1995 and 2015. Include units.

1.

2.

3.

4.

5.

7.

On July 1, 2019, Florida had a population of 21,477,737.

The area of Florida is 53,624.76 square miles. Find the population density of Florida.

Year

Population

1995

48,099

2015 c.

Population Density

136,996

How did the population density change between 1995 to 2015?

Unit 9: Two- and Three-Dimensional Shapes

61

2.

The Florida Fish and Wildlife Conservation Commission (FWC) tracks panthers in Florida. By tracking a panther, FWC can map the area in which a panther lives. The habitat of "Florida Panther 251" is shown with dimensions in kilometers, km.

a.

3.

Town A has 20 city blocks, and each city block 1 1 measures 20 of a mile by 2 mile. There are 6,500 people in this town. Find the population density of the town.

What is the area, in square kilometers (km2 ), of Florida Panther 251’s habitat?

At one time, the entire southeastern United States was the habitat of the Florida panther. Today, their habitat is restricted to several national parks and wildlife preserves in southwest Florida. Today, the total combined area of the Florida panther habitat is 9,466 𝑘𝑘𝑘𝑘2 . b.

62

Based on the area of Florida Panther 251’s habitat, how many Florida panthers could be living in southwest Florida?

Unit 9: Two- and Three-Dimensional Shapes

9.2.4 Guided Practice: Density Based on Area 1.

The number of leaves on a tree can be estimated using the crown of the tree. A tree crown is the top part of the tree where branches grow out from the main trunk.

The width of the tree’s crown is used as the diameter of the circle of the ground that lies under the tree’s crown.

The area where the leaves could fall is 4 times the area of the circle that lays under the tree’s crown. The approximate number of leaves per square foot is based on the average size of a leaf for each type of tree. Use the table to approximate the number of leaves that could fall from each tree.

Tree Sycamore Live Oak

Crown (in feet)

Approximate Number of Leaves per Square Foot

35

5

16

2.

The population density of Gainesville, Florida, is 2028.4 persons per square mile, and the population of Gainesville is 124,504. What is the area, in square miles, of Gainesville?

14

Unit 9: Two- and Three-Dimensional Shapes

63

3.

Wildlife biologists divided a habitat in the Bear Island area of the Big Cypress National Preserve into 96 squares, as shown. Each square has the dimensions of mile (mi) by

1 4

mi.

9.2.5 Wrap-Up: Reflection 1 4

1.

Complete the learning pyramid.

One question I would like to have answered is …

One mistake I learned from today is …

The biologists found that in the Bear Island habitat, there are 13.8 deer per square mile.

1 a. What is the population density for one of the mi by

b.

64

1 4

4

mi square?

One thing I am not sure about … YET!

One thing I am confident I know is …

How many deer live in the habitat area the biologists studied?

Unit 9: Two- and Three-Dimensional Shapes

Unit 9, Lesson 3: Cross Sections

9.3.1 Warm-Up: Patterns 1.

How many red squares are in the seventh pattern?

Benchmark MA.912.GR.4.1 Identify the shapes of two-dimensional cross-sections of three-dimensional figures.

Learning Target 

I can identify the shapes of two-dimensional cross sections of three-dimensional figures.

Unit 9: Two- and Three-Dimensional Shapes

65

9.3.2 Collaboration: Cross Sections

1.

What do you notice about the diagrams for horizontal slices?

2.

What do you notice about the diagrams for vertical slices?

3.

Discuss with your partner whether or not the shape of a cross section will change if the height of the plane is changed. Summarize your discussion.

4.

Complete the table by describing the shape of the cross section that is created when the figure is sliced by each.

The table below shows the resulting cross section for each type of slice.

Slice Figure Horizontal Plane Vertical Plane

Cylinder

Cone

Rectangular Prism

Figure Triangular Prism

Cylinder Cone Rectangular Prism

Sphere

Triangular Prism Sphere

66

Unit 9: Two- and Three-Dimensional Shapes

Cross Section Horizontal Shape

Vertical Shape

9.3.3 Guided Instruction: Drawing Cross Sections 1.

Draw the vertical cross section of the cone, given that 𝑊𝑊𝑊𝑊 = 6 units.

2.

A rectangular prism is shown with a horizontal cross section. The dimensions are given in centimeters (cm).

Draw the cross section on the grid and label its dimensions.

Unit 9: Two- and Three-Dimensional Shapes

67

3.

A sphere with radius of 4 inches (in.) with a horizontal cross section is shown.

Draw the cross section if it occurs the sphere’s center.

3 8

4.

A rectangular pyramid is shown, where the dimensions of the base are 12 centimeters (cm) by 21 cm and the pyramid’s height is 33 cm.

of the distance from

Draw the cross section if the horizontal plane is located at a height of 11 cm.

68

Unit 9: Two- and Three-Dimensional Shapes

3.

9.3.4 Your Turn!

1.

Draw the cross section formed by the intersection of a sphere with a diameter of 15 millimeters and a perpendicular plane passing through the center of the sphere.

2.

A cylinder has a radius of 4 inches (in.) and a height of 15 in. Draw the cross section formed by the intersection of a cylinder and a plane perpendicular to the base passing through the center of the cylinder.

Draw the cross section formed by the intersection of a pentagonal prism and a horizontal plane passing through the prism.

Unit 9: Two- and Three-Dimensional Shapes

69

Unit 9, Lesson 4: Solids of Rotation

9.3.5 Wrap-Up: Ticket Out the Door 1.

A hexagonal prism is shown. a.

Benchmark

Draw the cross section that results from a plane that is parallel to the base.

MA.912.GR.4.2 Identify three-dimensional objects generated by rotations of two-dimensional figures.

Learning Target 

b.

70

I can identify three-dimensional objects generated by rotations of two-dimensional figures.

Draw the cross section that results from a plane that is perpendicular to the base.

Unit 9: Two- and Three-Dimensional Shapes

9.4.1 Warm-Up: Notice and Wonder A Rubik’s Cube and a Slinky are shown.

9.4.2 Activity: Rotating Pencils Work with your partner to complete the following.

1.

Place a sticky note on the side of the pencil. If necessary, use tape to secure the sticky note.

2.

Make a prediction about which three-dimensional shape the sticky note would form when the pencil is rotated 360°.

Complete the table.

Prediction

1.

What do you notice?

2.

What do you wonder?

3.

Sketch the horizontal cross section of the Rubrik’s cube.

Sticky Note

3.

4.

Reasoning

Sketch the horizontal cross section of the Slinky.

4.

Using your pencil as a vertical axis of rotation, rotate the pencil 360°.

What three-dimensional figure did the sticky note seem to create?

Unit 9: Two- and Three-Dimensional Shapes

71

5.

Work with your partner to predict what threedimensional shape will be formed when each shape is rotated 360°.

Shape

Prediction

7.

Discuss with your partner whether the threedimensional shape would be different if the pencil were held horizontally instead of vertically. Summarize your discussion.

8.

Draw any four-sided, two-dimensional shape on a 3x5 card. Make it as unique as possible. Cut out your shape.

Draw the outline of your shape below.

Reasoning

Rectangle

Triangle

Semicircle

6.

Using your pencil as a vertical axis of rotation, rotate the pencil 360°. Then, sketch or describe the resulting three-dimensional shape.

Shape

Three-Dimensional Shape

Rectangle

Triangle

Semicircle

72

Unit 9: Two- and Three-Dimensional Shapes

9.

Sketch what you think the three-dimensional figure of your four-sided two-dimensional shape will look like if rotated around each axis.

Horizontal Axis

11. Discuss with your partner how drawing a reflection could help envision the three-dimensional figure.

Summarize your discussion.

Vertical Axis

12. Sketch what you think the three-dimensional figure of your partner’s four-sided, two-dimensional shape will look like if rotated around each axis.

10. Review your partner’s shape. Then, complete the following.

• Draw your partner’s shape on the grid below.

• Draw a horizontal axis of rotation.

• Draw the reflection of your partner’s shape.

Horizontal Axis

Vertical Axis

13. Compare your sketches with your partner. Discuss any differences and make any adjustments.

Unit 9: Two- and Three-Dimensional Shapes

73

9.4.3 Guided Instruction: Rotations With Dimensions 1.

Draw a two-dimensional figure and an axis of rotation that would result in the real-world object when the twodimensional figure is rotated around the axis of rotation.

Image

2.

Sketch the three-dimensional shape that is created.

Include any dimensions in units.

Two-Dimensional Shape ⃡ . 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 is rotated around 𝑊𝑊𝑊𝑊

Two-Dimensional Figure

A set of revolving doors with a radius of 3 feet. ∆𝐶𝐶𝐶𝐶𝐶𝐶 is rotated around ⃡𝑅𝑅𝑅𝑅.

An ice cream cone with a diameter of 4.5 centimeters (cm) and a height of 11 cm.

74

Unit 9: Two- and Three-Dimensional Shapes

ThreeDimensional Shape

9.4.4 Guided Practice: Rotations 1.

Describe and sketch the three-dimensional shape that is created when the rectangle is rotated around the 𝑦𝑦-axis. Include any known dimensions.

2.

Describe and sketch the three-dimensional shape that is created when the isosceles triangle is rotated around the 𝑦𝑦-axis. Include any known dimensions.

9.4.5 Wrap-Up: Ticket Out the Door 1.

Describe and sketch the three-dimensional shape that is created when the semicircle is rotated around its diameter. Include any known dimensions.

Unit 9: Two- and Three-Dimensional Shapes

75

Unit 9, Lesson 5: Cavalieri’s Principle

9.5.1 Warm-Up: Notice and Wonder Four different perspectives of the same stack of records are shown.

Benchmark MA.912.GR.4.5 Solve mathematical and real-world problems involving the volume of three-dimensional figures limited to cylinders, pyramids, prisms, cones, and spheres.

Learning Target 

76

I can use Cavalieri’s principle to give informal arguments about the formulas for the volumes of right and non-right cylinders, pyramids, prisms, and cones.

1.

What do you notice?

2.

What do you wonder?

Unit 9: Two- and Three-Dimensional Shapes

9.5.2 Collaboration: Exploring Cavalieri’s Principle Scenario 1

Consider the two stacks of pennies, where each stack contains the same number of pennies.

Discuss with your partner the relationship between the volumes of each stack.

1.

Scenario 2 Suppose two partners are both constructing towers that are to be 5 inches high. Partner A decided to build a tower using rectangles with areas of 16 square inches and partner B decided to construct a tower using squares with areas of 16 square inches.

The base of each partner’s tower is shown.

Summarize your discussion in the table below and have your partner initial that your summary is correct.

Summary

Partner’s Initials 2.

Discuss with your partner how the volumes will compare. Summarize your findings in the box below and have your partner initial that your summary is correct.

Summary

Unit 9: Two- and Three-Dimensional Shapes

Partner’s Initials

77

9.5.3 Guided Instruction: Cavalieri’s Principle

Scenario 3 ABC ice cream shop decided to run a special for the week and they’re offering two serving options for you to choose from. Option 1 is a cone with a circular base that has an area of 4𝜋𝜋 square inches and a height of 4 inches. Option 2 is a triangular pyramid container with a base of 4𝜋𝜋 square inches and a height of 4 inches. The base of each option is shown.

3.

1.

Using Cavalieri’s Principle, determine whether or not the following solids have the same volumes.

Discuss with your partner how the volumes will compare to one another. Summarize your findings in the box below and have your partner initial that your summary is correct.

Summary

78

Cavalieri’s Principle for Three-Dimensional Solids If every plane parallel to the two bases of two solids results in cross sections of equal area and the two solids have congruent altitudes, then the solids have equal volumes.

Partner’s Initials

Unit 9: Two- and Three-Dimensional Shapes

2.

An oblique prism is shown.

9.5.4 Guided Instruction: Slant Height vs.

Vertical Height A square pyramid is shown with measurements in millimeters (mm).

a.

b.

c.

d.

What shape is each cross-section of the oblique prism?

What is the area formula for the base?

What is the volume formula for an oblique rectangular prism?

5 mm

3 mm

𝑥𝑥 mm

3 mm

Samantha and Julian each found the measure of the slant height, 𝑥𝑥, by first drawing a diagram and then applying the Pythagorean Theorem. Their work is shown.

Samantha

Julian

𝑐𝑐 2 = 32 + 52

𝑐𝑐 2 = 1.52 + 52

Discuss with your partner how the formula for the oblique rectangular prism compares to the formula for a right rectangular prism. Explain your reasoning.

Unit 9: Two- and Three-Dimensional Shapes

79

1.

Discuss with your partner which student correctly set up their work to determine the slant height. Summarize your discussion.

9.5.5 Guided Practice: Cavalieri’s Principle in the Real World 1.

The slant height of an object is the distance from the base to the apex along a lateral face, whereas the vertical height or altitude of an object is the perpendicular distance from the apex to the base.

2.

80

Consider the stack of sticky note pads below.

If each pad is a square with side lengths of 3 inches and a height of 0.97 inches, approximate the volume of a stack of 8 sticky note pads.

Discuss with your partner which height is important when finding the volume of an oblique figure.

Summarize your discussion.

Unit 9: Two- and Three-Dimensional Shapes

2.

Two stacks of 12 chocolate squares are shown.

Stack A

5

of each stack.

Stack A

While cleaning out an attic, you found an old stack of compact discs (CDs) that were used to store music. If each disc has an area of 36𝜋𝜋 cm2, explain how you can find the approximate volume of a stack of 25 CDs.

4.

A zigzag crystal vase is shown, where the vase has a height of 10 inches (in.). The cross-sections that are parallel to the base are always rectangles that are 6 in. wide by 3 in. long.

Stack B

5

Each chocolate square has the dimensions 1 inches (in.) by 1 8 in. Each square is

3.

1 8

8

in thick. Find the volume

a.

Determine the volume of the vase, assuming the crystal itself has no thickness.

b.

The crystal is actually half an inch thick on each of the sides and on the bottom. Approximately how much space is contained within the vase? Explain or show your reasoning.

Stack B

Unit 9: Two- and Three-Dimensional Shapes

81

Unit 9, Lesson 6: Finding Volume

9.5.6 Wrap-Up: Reflection Complete each statement based on your understanding up to this point.

Benchmark 1.

I am still unsure about __________________________

MA.912.GR.4.5 Solve mathematical and real-world problems involving the volume of three-dimensional figures limited to cylinders, pyramids, prisms, cones, and spheres.

_______________________________________________ Learning Target 2.

One question that I have that will help me better understand is _________________________________



I can find the volume of spheres, prisms, cones, pyramids, and cylinders.

_______________________________________________

82

Unit 9: Two- and Three-Dimensional Shapes

9.6.1 Warm-Up: Would You Rather?

1.

Would you rather have a pool that is 20 meters (m) by 8 m by 6 m, or a pool that is 1550 centimeters (cm) by 650 cm by 725 cm? a.

Use mathematics to compare the two pools.

b.

Which pool would you rather have?

9.6.2 Guided Instruction: Finding Volume 1.

A cone and cylinder are shown, where the radius of the base of each is 5 units and the height of each is 8 units.

Cone

Cylinder

a.

Suppose the cone is filled with water. Discuss with your partner how many cones you think you would need to fill the cylinder with water. Write your estimate.

b.

Given the volume of a cylinder is 𝑉𝑉 = 𝐵𝐵ℎ, discuss with your partner how to re-write the formula to find the volume of the cone.

c.

The volume formula for a cone is 𝑉𝑉 = _____________

A cone is a three-dimensional figure with a circular base and an apex that is connected to the base by a collection of line segments that form a curved surface.

Unit 9: Two- and Three-Dimensional Shapes

83

2.

A cone is shown with a diameter of 16 millimeters (mm), a height of 22 mm, and a slant height of 23.4 mm.

3.

The bases of the rectangular pyramid and rectangular prism shown have congruent bases, and each has the same height.

Determine the volume of the cone, in terms of 𝜋𝜋.

84

a.

Suppose the rectangular pyramid is filled with water. Discuss with your partner how many rectangular pyramids you think you would need to fill the rectangular prism with water. Write your estimate.

b.

Given the volume of a prism is 𝑉𝑉 = 𝐵𝐵 ∙ ℎ, discuss with your partner how to re-write the formula to find the volume of the pyramid.

c.

The volume formula for a regular pyramid is 𝑉𝑉 = _____________

Unit 9: Two- and Three-Dimensional Shapes

A pyramid is a figure with a polygonal base and triangular faces. The triangular faces have the same size and shape and they connect the sides of the base to a common point called the apex.

4.

A pyramid is shown, whose base is a rectangle with dimensions of 10 centimeters (cm) by 14 cm. The height of the pyramid is 17 cm. Determine the volume of the pyramid.

A sphere is a three-dimensional figure, where all points are equidistant from a point called the center.

A sphere with center 𝐿𝐿 is shown, where ̅​̅​̅​̅​̅ 𝐿𝐿𝐿𝐿 is a radius ̅​̅​̅​̅ is a diameter. The and 𝑇𝑇𝑇𝑇 circle that passes through the points 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 is known as a great circle since it is the largest possible circle that can be made around the sphere.

The volume formula for a 4 sphere is 𝑉𝑉 = 3 𝜋𝜋𝑟𝑟 3 .

5.

Find the volume of the sphere 𝐺𝐺, where 𝐺𝐺𝐺𝐺 = 3 using 𝜋𝜋 ≈ 3.14. Round to the nearest thousandth.

Unit 9: Two- and Three-Dimensional Shapes

85

6.

Composite figures are made up of more than one figure. The composite figure shown is a cone on top of a cylinder.

Check all that must be true about the composite figure.     

7.

86

8.

A composite figure is shown, where 𝐻𝐻𝐻𝐻 = 3.2, 𝐾𝐾𝐾𝐾 = 4.8, 𝐷𝐷𝐷𝐷 = 10.7, and 𝑊𝑊𝑊𝑊 = 11.1. 𝑊𝑊𝑊𝑊 is the height of the pyramid. Find the volume of the composite figure.

The heights have the same measure.

The diameters have the same measure.

The radii have the same measure.

The bases have the same area.

The volume of this composite figure can be found using the formula 𝑉𝑉 = 2𝜋𝜋𝑟𝑟 2 ℎ.

Discuss with your partner the statements that were false. Rewrite each false statement so that it would be true.

Unit 9: Two- and Three-Dimensional Shapes

9.6.3 Guided Practice: Finding Volume

9.6.4 Your Turn!

1.

Find the volume of a hemisphere that has a diameter of 9.4 inches.

1.

Find the volume of a cylinder with a diameter of 18.3 millimeters (mm) and a height of 6.4 mm. Round your answer to the nearest thousandth.

2.

A composite figure consists of a hemisphere and a cone. The base of the cone and the great circle of the hemisphere have the same diameter. The radius of the 3 hemisphere is 4 centimeters (cm). The height of the

2.

Find the volume of a pyramid whose height is 15.7 inches (in.), and whose base is a rectangle with dimensions of 7.6 in. by 12.4 in.

cone is 9

1

12

4

centimeters (cm). Find the volume of the

composite figure using

22 7

for 𝜋𝜋.

Unit 9: Two- and Three-Dimensional Shapes

87

Unit 9, Lesson 7: Volume in the Real World

9.6.5 Wrap-Up: Ticket Out the Door

1.

Find the volume of the composite figure if 𝑀𝑀𝑀𝑀 = 17, 𝐷𝐷𝐷𝐷 = 8.5, and 𝐾𝐾𝐾𝐾 = 26. Round your answer to the nearest thousandth.

Benchmark MA.912.GR.4.5 Solve mathematical and real-world problems involving the volume of three-dimensional figures limited to cylinders, pyramids, prisms, cones and spheres.

Learning Target 

88

I can use volume to solve problems with real-world applications.

Unit 9: Two- and Three-Dimensional Shapes

9.7.1 Warm-Up: Math Fail

What do you notice about the sign that makes this a math fail?

Josiah works for a sporting goods company in the shipping department. Josiah is trying to determine the best solution for packing a basketball with a diameter of 28.5 inches. a.

What is the volume of the basketball?

b.

What size box do you think would be the best fit?

c.

Find the volume of the three options of boxes below, from which Josiah can choose.

Rectangular Prism

a.

1.

30 in

The clearance sign shown is what is known as a “math fail.”

Write a letter to the company explaining why this is a math fail and how to fix the issue.

Cylinder

28.5 in

Volume =__________

b.

29 in

30 in

1.

9.7.2 Collaboration: Packing Peanuts

Volume =__________ Cube

Unit 9: Two- and Three-Dimensional Shapes

29 in

Volume =__________

89

Discuss with your partner how to determine the amount of space left over in the package after the basketball has been placed in it. Summarize your discussion.

9.7.3 Guided Practice: Volume 1.

Keon is building a raised garden bed that will have a width of 3.5 feet (ft), a depth of 2 ft, and length of 3 ft.

If Keon fills the garden bed with soil, how much soil does he need to purchase?

2.

Jasper wants to buy a new coffee mug, and he has narrowed it down to two options. The dimensions are given in inches (in.)

5.25 in

Option A

e.

90

After the basketball has been placed in the package, the remaining space is filled with packing peanuts. If each peanut is 1 cm3 , which box will use the least amount of packing peanuts?

Diameter: 2 in

Option B

5 in

d.

Radius: 1.25 in

a.

What is the volume, rounded to the thousandths place, of option A?

b.

What is the volume, rounded to the thousandths place, of option B?

Unit 9: Two- and Three-Dimensional Shapes

2.

9.7.4 Your Turn!

1.

A tennis ball canister with three tennis balls is shown.

The diameter of the canister is 2.7 inches (in.) and the height is 8.1 in. a.

What is the volume of the canister?

b.

What is the volume of ͳ tennis ball?

Club Cool in Epcot offers free samples of Coke products that are distributed worldwide. Flavors include Fanta Pineapple, Fanta Melon, Inca Cola, VegitaBeta, and Sparberry. You can choose to have your sample served in either of the below containers shown.

Gian, Yanessa, and Jack made different conclusions about the drink containers. Who is correct and why?

Gian: The volumes are the same because the height of the cone is 4 times its radius, and both containers have the same radius.

c.

How much air is left in the canister after the tennis balls have been placed inside?

Yanessa: The volume of the sphere is greater than the 4 1 cone’s because 3 𝜋𝜋𝑟𝑟 3 is greater than 3 𝜋𝜋𝑟𝑟 2 .

Jack: The volume of the cone is greater than the sphere’s because the height is 4 times the radius of both containers.

Unit 9: Two- and Three-Dimensional Shapes

91

3.

Julian owns a moving company and has a truck that measures 15 ft × 7.5 ft × 7.5 ft.

4.

The Pyramid of Cestius, pictured below, is located in Rome, Italy. The base of the pyramid is a square with side lengths of 30 meters and stands at a height of 36.5 meters. What is the volume of the pyramid?

He charges individuals by the number of boxes in the moving truck. If each box is a cube with side lengths of 1.5 ft, how many boxes can he fit in the truck?

92

Unit 9: Two- and Three-Dimensional Shapes

Unit 9, Lesson 8: Surface Area of Three-Dimensional Figures

9.7.5 Wrap-Up: Reflection 1.

Take a SELFIE by checking the box that best matches what you did today.

Showed my work Explained my answers Lots of math vocabulary used

Benchmark MA.912.GR.4.6 Solve mathematical and real-world problems involving the surface area of three-dimensional figures limited to cylinders, pyramids, prisms, cones, and spheres.

Found multiple solutions I persevered through the problem Eliminated careless errors

Learning Targets 



I can solve mathematical problems involving the surface area of cylinders, pyramids, prisms, cones, and spheres.

I can solve mathematical problems involving the lateral area of cylinders, pyramids, prisms, cones, and spheres.

Unit 9: Two- and Three-Dimensional Shapes

93

9.8.2 Collaboration: Using Nets to Find Surface Area

9.8.1 Warm-Up: Which One Doesn’t Belong?

1.

Which one doesn’t belong? Explain your answer.

1.

The lateral area of a three-dimensional figure is the sum of the area of the figure’s faces.

The surface area of a three-dimensional figure is the sum of the lateral area of the figure’s faces and the area of the figure’s bases.

Work with your partner to determine a formula for the lateral area and the surface area of the figure you have been given. Complete the table.

Our figure was:

Dimension Variable Used

94

Unit 9: Two- and Three-Dimensional Shapes

Triangular Prism

9.8.3 Activity: Gallery Walk 1.

During the gallery walk, record the formulas. Make any useful notes on each figure’s net.

Figure Net Cylinder

Lateral Area = Surface Area =

Lateral Area =

Surface Area =

Square Pyramid Lateral Area =

Cube Lateral Area =

Surface Area = Surface Area =

Rectangular Prism Lateral Area =

Rectangular Pyramid

The height of the triangle that is a face of a pyramid is also known as slant height, ℎ𝑠𝑠 .

Lateral Area = Surface Area =

Surface Area =

The height of the triangle that is a face of a pyramid is also known as slant height, ℎ𝑠𝑠 .

Unit 9: Two- and Three-Dimensional Shapes

95

9.8.4 Bring It Together: Formulas

2.

There are many ways to write and to understand formulas. Discuss with your partner which formulas make more sense to you. Discuss how the formulas in the formula chart can be used to help you remember how to find surface area. Summarize your discussion.

3.

The chart does not have a formula for lateral area. For each of the figures, except for spheres, circle the part of the surface area formula that gives the lateral area.

Using the net of three-dimensional objects can help to find the lateral area and surface area. The net of a cone and a sphere make deriving the formulas for surface area from the net more challenging. A chart of formulas is given, where 𝑆𝑆𝑆𝑆 represents surface area.

Figure

Prism Cylinder Cone Regular Pyramid (a pyramid whose base is a regular polygon) Sphere 1.

96

Surface Area

𝑆𝑆𝑆𝑆 = 2𝐵𝐵 + 𝑃𝑃ℎ 𝑆𝑆𝑆𝑆 = 2𝐵𝐵 + 𝐶𝐶ℎ Or 𝑆𝑆𝑆𝑆 = 2𝐵𝐵 + 𝑃𝑃ℎ 𝑆𝑆𝑆𝑆 = 𝐵𝐵 + 𝜋𝜋𝜋𝜋ℎ

Variables

𝐵𝐵 ℎ 𝐵𝐵 𝐶𝐶 𝑃𝑃 ℎ 𝐵𝐵 ℎ

= base, 𝑃𝑃 = perimeter, = height = base, = circumference (or = perimeter, = height) = base, 𝑟𝑟 = radius, = height

1 𝐵𝐵 = base, 𝑃𝑃 = perimeter, 𝑆𝑆𝑆𝑆 = 𝐵𝐵 + 𝑃𝑃ℎ𝑠𝑠 ℎ𝑠𝑠 = slant height 2 𝑆𝑆𝑆𝑆 = 4𝜋𝜋𝑟𝑟 2

𝑟𝑟 = radius

The formulas in the table may not match the formulas recorded during the gallery walk. Which figures have formulas that are different?

Unit 9: Two- and Three-Dimensional Shapes

9.8.5 Guided Instruction: Lateral Area and Surface Area

1. A square pyramid has a slant height of 7 inches, and the area of the base is 16 square inches. What is the surface area, in square inches?

3.

If a cylinder has a height of 16 centimeters and a radius of 8 centimeters. a.

Find the lateral area. Leave your answer in 𝜋𝜋 form.

b.

Find the surface area. Leave your answer in 𝜋𝜋 form.

2.

The surface area of a sphere is 144𝜋𝜋 feet. What is the radius of the sphere?

4. A right triangular prism has base side lengths of 8, 15, and 17 feet (ft), where 17 ft is the hypotenuse of the triangle. The prism has a height of 7 ft. What is the surface area, in square feet, of the triangular prism?

Unit 9: Two- and Three-Dimensional Shapes

97

9.8.6 Guided Practice: Lateral Area and Surface Area 1.

A pyramid has a square base whose area is 49 square feet with a height of 11.5 feet. Find the lateral area, in square feet, of the pyramid.

2.

A rectangular prism has a length of 12.5 inches (in), a width of 8 in., and a height of 5 in. Determine the surface area, in square inches, of the rectangular prism.

98

3.

The radius of a sphere is 18.7 feet. Find, in square feet, the surface area of the sphere. Round your answer to thousandths place.

Unit 9: Two- and Three-Dimensional Shapes

9.8.7 Your Turn!

1.

9.8.8 Wrap-Up: Error Analysis

Find the lateral area, in square meters, of a cone that has a radius of 3 meters (m) and a slant height of 8.5 m. Round your answer to thousandths place.

1.

Cody and Mila are finding the surface area of a pyramid whose base area is 144 inches and slant height is 16 inches. Their work is shown in the table.

Cody’s Work 1 𝑆𝑆𝑆𝑆 = 𝐵𝐵 + 𝑃𝑃ℎ𝑠𝑠 2 1 𝑆𝑆𝑆𝑆 = 144 + (48)(16) 2 𝑆𝑆𝑆𝑆 = 528 square inches

2.

Mila’s Work 1 𝑆𝑆𝑆𝑆 = 𝐵𝐵 + 𝑃𝑃ℎ𝑠𝑠 2 1 𝑆𝑆𝑆𝑆 = (144) + (144)(16) 2 𝑆𝑆𝑆𝑆 = 1296 square inches

a.

Whose work is correct?

b.

Write a note to the student whose work is incorrect describing the mistake the student made.

A cylinder has a radius of 5 inches (in) and a height of 3.75 in. Find the surface area, in inches, of the cylinder.

Round your answer to thousandths place.

Unit 9: Two- and Three-Dimensional Shapes

99

Unit 9, Lesson 9: Surface Area in the Real World

Benchmark MA.912.GR.4.6 Solve mathematical and real-world problems involving the surface area of three-dimensional figures limited to cylinders, pyramids, prisms, cones, and spheres.

9.9.1 Warm-Up: Math Magic Arthur Benjamin (b. 1961) is a professor of mathematics at Harvey Mudd College in California. Dr. Benjamin is often referred to as a “mathemagician” because he is known for his quick mental computational skills. Dr. Benjamin’s TED talks have been viewed millions of times on the internet.

Try one of Dr. Benjamin’s math “magic tricks.”

• • • • •

Learning Targets 



I can solve real-world problems involving the surface area of cylinders, pyramids, prisms, cones, and spheres.

I can solve real-world problems involving the surface area of composite figures.

100

Pick a number between 1 and 10.

Double your number.

Add 10 to your number.

Divide by 2.

Subtract the number you started with originally.

1.

What number did you get?

2.

Compare your number with your partner’s number.

What do you notice?

3.

Why do you think the “trick” works?

Unit 9: Two- and Three-Dimensional Shapes

9.9.2 Collaboration: The Best Package Manuel is a graphic designer and works with various types of packaging. For the artwork, Manuel needs a container with a surface area of at least 250 square inches (in.) to ship the company’s products.

1.

The three types of packaging that the company has in stock are shown in the table. Find the surface area for each type of packaging.

Packaging Dimensions Rectangular Prism 𝑙𝑙: 7 in. 𝑤𝑤: 10 in. ℎ: 4 in.

Cylinder

3.

Packaging Rectangular Prism

Cost of Packaging

Cylinder $2.25 per 100 square feet.

Surface Area

Cube $1.25 per 100 square feet.

𝑑𝑑: 8 in. ℎ: 8 in.

Cube

Pricing $1.75 per 100 square feet.

4.

2.

The prices for packaging are listed in the table. Based on the surface area of each package, find the cost of each type of packaging.

𝑠𝑠: 12 in.

If Manuel wants to use the cheapest packaging possible for his client, which packaging should he choose?

Rank the packaging’s surface area from smallest to largest.

Rank

Packaging

1 (smallest) 2

3 (largest)

Unit 9: Two- and Three-Dimensional Shapes

101

9.9.3 Guided Instruction: Real-Life Applications of Surface Area

2.

Jeni makes a two-layer cake, as shown, where the measurements are in inches (in.) She will frost all of the cake except for the base. The frosting between the pyramid and the prism helps to hold the two cakes together. Determine the total surface area that Jeni needs to frost.

3.

Jeni’s recipe for buttercream icing typically frosts an 8-inch round cake that is 1.5 inches high. How many batches of frosting will Jeni need to frost the two-layer cake?

Jeni owns a bakery and is making two cakes.

Cake A Cake A is in the shape of a rectangular prism that is 3 inches thick, 9 inches long, and 12 inches wide.

1.

Cake B The cake is in the shape of a cylinder and has a diameter of 8 inches and a height of 6 inches.

When Jeni frosts the cake, she applies the frosting to all surfaces except for the bottom of the cake. Determine the amount of frosting Jeni needs to apply to each cake. Show your work.

Cake A

Cake B

8

102

Unit 9: Two- and Three-Dimensional Shapes

9.9.4 Guided Practice: Real-Life Applications of Surface Area 1.

Lourdes’ garage is in the shape of a square pyramid.

The garage is shown, with dimensions given in feet (ft).

2.

An architect is designing an art gallery. She used computer software to create a rough sketch of her idea.

Her sketch is shown with measurements in meters (m).

She would like for the art gallery to be all glass. To determine the estimated cost, she would need to find the area of all of the surfaces except for the floor.

Determine the total surface area, in square meters, of the glass in the art gallery. Round to the nearest thousandth.

18 ft

Lourdes is replacing the roof and installing shingles. In roofing terminology, a "square" is 100 square feet.

1 Composition shingles come in bundles that cover 3 of a

square.

How many bundles should Lourdes buy to cover the roof?

Unit 9: Two- and Three-Dimensional Shapes

103

Unit 9, Lesson 10: Determining How Dilations Affect Area

9.9.5 Wrap-Up: Ticket Out the Door 1.

Jillian was building a birdhouse. Some of the dimensions, in inches (in.), of the birdhouse are shown.

Determine the total amount of wood, in square inches, that Jillian will need to build the outside of the birdhouse.

Benchmark MA.912.GR.4.3 Extend previous understanding of scale drawings and scale factors to determine how dilations affect the area of two-dimensional figures and the surface area or volume of three-dimensional figures.

Learning Target 

104

I can discover the relationship between area and the scale factor of a dilation.

Unit 9: Two- and Three-Dimensional Shapes

9.10.2 Exploration: Scale Factor and Area

9.10.1 Warm-Up: Where Is Here?

1.

A picture of a directional sign is shown.

1.

2.

Square 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is shown, where 𝑊𝑊𝑊𝑊 = 1 unit.

a.

Determine the area of 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴.

b.

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is dilated using point 𝐴𝐴 as the center of dilation using each of the given scale factors.

Complete the table.

What do you notice about the sign?

Where do you think the picture was taken?

Scale Factor

3.

The Earth’s average polar circumference is 24,859 miles. What do you think the sign means?

Side Length of Dilation

1 2

3 4

2

3

4

Area of Dilated Square

Unit 9: Two- and Three-Dimensional Shapes

105

c.

Write a conjecture about the relationship between the length of the scale factor and the value of the area.

b.

Share your conjecture with your partner. Listen to your partner’s conjecture and write a summary. Have your partner initial that it is summarized correctly.

Summary of Partner’s Conjecture

d.

2.

Determine the area of 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 if 𝑊𝑊𝑊𝑊 = 4 units.

Suppose square 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 is dilated using point 𝐷𝐷 as the center of dilation for each of the given scale factors.

Complete the table for 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 given that 𝑇𝑇𝑇𝑇 = 4 units.

Scale Factor Side Length of Dilation Area of Dilated Square a.

106

Partner’s Initials

1 2

3 4

2

3

4

c.

Review your conjecture and make any revisions, if needed.

Review your conjecture from the previous problem and make any revisions, if needed.

Unit 9: Two- and Three-Dimensional Shapes

3.

Rectangle 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 is shown.

a.

4.

Graph 𝑆𝑆′𝐵𝐵′𝐾𝐾′𝐷𝐷′ on the coordinate plane.

Using the coordinate grid, find the area of the rectangle.

Area=_____ square units

b.

Dilate 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 by a scale factor of 3 centered at the origin. Write the new coordinates in the table.

Point

Preimage

𝑆𝑆

(−4,4)

𝐾𝐾

(3,1)

𝐵𝐵

𝐷𝐷

(3,4)

(−4,1)

a.

The area of rectangle 𝑆𝑆′𝐵𝐵′𝐾𝐾′𝐷𝐷′ is _____ square units.

b.

Does the conjecture you wrote about the relationship being area and scale factor when dilating a square hold true for rectangle 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆?

Image

Unit 9: Two- and Three-Dimensional Shapes

107

5.

Parallelogram 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 is shown.

d.

What is the relationship between the area of 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 and 𝐹𝐹′𝑁𝑁′𝐶𝐶′𝑇𝑇?

e.

How can the scale factor of

2 3

be used to determine

the area of 𝐹𝐹′𝑁𝑁′𝐶𝐶′𝑇𝑇′ from the area of 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹? a.

What is the area of 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹, in square units?

b.

Dilate 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 by a scale factor of

2 3

centered at the

origin. Write the coordinates of the preimage and image in the table.

Point

Preimage

Image

𝐹𝐹

𝑁𝑁 𝐶𝐶 𝑇𝑇

c.

108

Plot the coordinates of 𝐹𝐹′𝑁𝑁′𝐶𝐶′𝑇𝑇′ on the same coordinate plane as 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹.

Unit 9: Two- and Three-Dimensional Shapes

9.10.3 Guided Practice: Solving Problems Involving Scale Factor and Area 1.

Trapezoid 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 is shown on the coordinate plane.

b.

Point

Find the coordinates of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 given the scale factors in the table. Use the preimage coordinates to find the coordinates of the images. Both dilations are centered at (0,0).

Preimage

𝑀𝑀

𝟏𝟏

Dilation 𝒌𝒌 = 𝟔𝟔 (𝑴𝑴′ 𝑻𝑻′ 𝑹𝑹′ 𝑫𝑫′)

𝟒𝟒

Dilation 𝒌𝒌 = 𝟑𝟑 (𝑴𝑴"𝑻𝑻"𝑹𝑹"𝑫𝑫")

𝑇𝑇

𝑅𝑅

𝐷𝐷

a.

c.

Find the area of 𝑀𝑀′𝑇𝑇′𝑅𝑅′𝐷𝐷′.

d.

What effects does the scale factor of

Find the area of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀.

1 𝐴𝐴 = (______ + ______)(_____) 2

The area of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 is _____ square units.

original area?

Unit 9: Two- and Three-Dimensional Shapes

𝟏𝟏 𝟔𝟔

have on the

109

e.

Find the area of 𝑀𝑀′′𝑇𝑇′′𝑅𝑅′′𝐷𝐷′′.

9.10.4 Your Turn!

1.

f.

What effects does the scale factor of original area?

g.

𝟒𝟒 𝟑𝟑

Circle 𝑆𝑆 is shown with a radius of √20.

have on the

How does the area of the trapezoid with a scale 1 factor of 6 differ from the area of the trapezoid with 4

a scale factor of 3?

a.

110

Dilate circle 𝑆𝑆 using a scale factor of 1.5 centered at the origin, and graph the resulting image on the coordinate plane.

Unit 9: Two- and Three-Dimensional Shapes

b.

Find the area of the circles.

Area of Preimage

Area of Image

9.10.5 Wrap-Up: Growth Mindset Question?

Did I use the exploration as a learning opportunity?

Did I use strategies to ‘un-stick’ myself when I found the activity difficult?

Statement

My Description o o o o

Never Sometimes Usually Always

o Never

o Sometimes o Usually o Always

My Description

Today I feel like I did a good job with ….

Today I felt confused by …

Unit 9: Two- and Three-Dimensional Shapes

111

Unit 9, Lesson 11: Determining Surface Area Given Dimension Changes

9.11.1 Warm-Up: Bell Work 1.

Trapezoid 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 is shown in the coordinate plane.

Benchmark MA.912.GR.4.3 Extend previous understanding of scale drawings and scale factors to determine how dilations affect the area of two-dimensional figures and the surface area or volume of three-dimensional figures.

Learning Targets  

I can find the area of a figure when one or both dimensions are changed.

I can find the surface area of a three-dimensional figure when one or both dimensions are changed.

a.

Find the area of 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷.

b.

Find the coordinates of the image of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 given the scale factor in the table. Use the preimage coordinates to find the coordinates of the image.

The dilation is centered at (0,0).

Point

Preimage

𝐷𝐷 𝑃𝑃 𝐶𝐶 𝐿𝐿

112

Unit 9: Two- and Three-Dimensional Shapes

Dilation 𝒌𝒌 = 𝟔𝟔 (𝑫𝑫′ 𝑷𝑷′ 𝑪𝑪′𝑳𝑳′)

9.11.2 Exploration: Ryan’s Renovations

3.

Ryan is building a new house and wants to make the size of the kitchen larger. The current kitchen is 24 feet wide and 10 feet long. Ryan wants to make the new kitchen 20 feet long but leave the width the same.

1.

Pool 1

Pool 2

Pool 3

Find the area of the original kitchen and the area of the new kitchen.

Area of the Original Kitchen

2.

Find the surface area of each of the shapes for the three pool options. Round to the nearest thousandth.

Area of the New Kitchen

The construction company is giving Ryan more options for building a swimming pool. The three options are listed in the table.

How do the areas compare?

Ryan also wants to install a pool for his new house. The construction company gives Ryan three shape options for the new pool. Each pool’s dimensions are given in feet (ft).

Pool 1 Cylindrical diameter = 16 ft height= 12 ft

Pool 2 Rectangular Prism length= 30 ft width= 20 ft height= 12 ft

Pool 3 Cube Each side= 15 ft

Pool 4 Cylindrical diameter = 8 ft height = 6 ft 4.

Pool 5 Pool 6 Rectangular Prism Cube length = 15 ft Each side = 7.5 ft width = 10 ft height = 6 ft

Describe how the dimensions of pair of pools are related.

Pair Description Pool 1 and Pool 4 Pool 2 and Pool 5 Pool 3 and Pool 6

Unit 9: Two- and Three-Dimensional Shapes

113

5.

Find the surface area of each of the shapes for the three new options for a pool. Round to the nearest thousandth.

Pool 4

Pool 5

The cost of building the swimming pool is dependent upon the amount of material. Since the top of each of the shapes would be water, determine the surface area of each pool that could be used to determine the cost.

Pool 6 7.

Complete the table.

Pool

Surface Area for Material

1 2 3 4 6.

5

Record the surface area of all pool shapes in the table below.

Pool 1

6

Surface Area 8.

If Ryan wants the least expensive pool, which one should he purchase?

9.

Which pool would you purchase? State your reasoning.

2 3 4 5 6

114

Unit 9: Two- and Three-Dimensional Shapes

9.11.3 Guided Practice: Ryan’s House

2.

Ryan is building a bathroom and needs to install either a bathtub or a shower. The options for each bathroom size are given.

Shower 1 length = 4 ft width = 6 ft height = 12 ft 1.

Ryan determines that the dimensions of the shower and tub are not up to code. The updated dimensions are shown. Use the new dimensions to determine the surface area of each room.

Shower 2 length = 6 ft width = 9 ft height = 12 f

Tub 1 length = 4 ft width = 8 ft height = 10 ft

Surface Area

Tub 2 length = 6 ft width = 12 ft height = 15 ft

Surface Area

In order to determine the amount of drywall, Ryan needs to find the surface area of each of the room options.

Shower 1

Tub 1

Two of Shower 2’s dimensions were changed by a factor of 1.5, whereas all of Tub 2’s dimensions were changed by a factor of 1.5. Discuss with your partner on how to use the factor 1.5 and the number of changes in dimensions to find the new surface area from the original surface area.

3.

Complete the statements.

__________ × surface area of Shower 1 = surface area of Shower 2 __________ × surface area of Tub 1 = surface area of Tub 2

Unit 9: Two- and Three-Dimensional Shapes

115

9.11.4 Your Turn!

9.11.5 Wrap-Up: Reflection

The living room of Ryan’s house is 12 feet long and 18 feet wide.

1.

1.

The room is a rectangular shape. Find the area of Ryan’s living room.

Use the emoji scale to describe how well you understand how to find the area of a figure if one or both of the dimensions are changed.

2.

3.

How would the area of the living room change if Ryan decided to enlarge the dimensions by 25%?

What is the scale factor for a dimension increase of 25%?

2.

Use the emoji scale to describe how well you understand how to find the surface area of a threedimensional figure when one or both dimensions are changed.

How did the area of the living room change with the new dimensions?

116

4.

Unit 9: Two- and Three-Dimensional Shapes

Unit 10 - Arcs and Angle Relationships in Circles Lesson 10.1 Sectors 10.2 Arc Length 10.3 Radian Measure 10.4 Solving Problems Involving Arc Length and Area of a Sector 10.5 Solving Problems Involving Arc Measure and Central Angles 10.6 Solving Problems Involving Inscribed Angles 10.7 Solving Problems Involving Intersecting Chords 10.8 Solving Problems Involving Intersection of Secants 10.9 Solving Problems Involving Tangents 10.H1 Constructing Tangent Lines

Unit 10: Arcs and Angle Relationships in Circles

Page Number 119 124 130 135 141 148 153 159 164 169

117

The following B.E.S.T. Standards will be covered in this unit: MA.912.GR.6.2 Solve mathematical and real-world problems involving the measures of arcs and related angles.

• Clarification 1: Within the Geometry course, problems are limited to relationships between inscribed angles; central angles; and angles formed by the following intersections: a tangent and a secant through the center, two tangents, and a chord and its perpendicular bisector.

MA.912.GR.6.4 Solve mathematical and real-world problems involving the arc length and area of a sector in a given circle.

• Clarification 1: Instruction focuses on the conceptual understanding that the length of the arc intercepted by an angle is proportional to the radius.

MA.912.GR.5.5 (Honors only) Given a point outside a circle, construct a line tangent to the circle that passes through the given point.

• Clarification 1: When given a circle, the center must be provided.

• Clarification 2: Instruction includes using a compass and straightedge, string, reflective devices, paper folding or dynamic geometric software.

118

Unit 10: Arcs and Angle Relationships in Circles

Benchmark

MA.912.GR.6.4 Solve mathematical and real-world problems involving the arc length and area of a sector in a given circle.

Learning Targets  

10.1.1 Warm-Up: Pac-Man

Unit 10, Lesson 1: Sectors

I can define a sector of a circle.

I can explain how the area of a sector is defined as a fractional part of the area of a circle.

The classic and popular Pac-Man video game was released in Japan in May of 1980 and was in the United States by October of that same year. The yellow, pie-shaped character travels around a maze trying to eat dots while avoiding the ghosts hunting him.

Watch the video of Pac-Man going around the maze and eating dots.

1.

What do you notice about Pac-Man’s body as he opens and closes his mouth to eat the dots?

2.

What do you wonder as you watch Pac-Man move around the maze?

Unit 10: Arcs and Angle Relationships in Circles

119

10.1.2 Exploration: Fractions of a Circle

2.

The point where a pizza is divided to create precisely equal slices is a central angle.

A central angle is an angle with its vertex at the center of a circle with radii as its side.

Work with your partner to complete the following.

1.

Pizza sizes are given in terms of the pizza’s diameter. The standard size for most large circular pizzas is 14 inches (in.) and it is usually cut into 10 triangular slices.

a.

What is the approximate area of a 14-inch pizza? Use 22 for 𝜋𝜋.

7

b.

If the slices are cut precisely and equally, what fraction and percentage of the pizza is 5 slices?

Fraction

Percentage

𝜽𝜽

The central angle is a fraction of the total 360° in a circle. a.

c.

Complete the table. The first row of the table has been filled in.

What is the area of 5 slices?

Fraction Half of a Circle

d.

What is the area of 1 slice?

Quarter of a Circle Third of a Circle Tenth of a Circle

120

Unit 10: Arcs and Angle Relationships in Circles

1⁄2

Central Angle in Degrees (°)

(1⁄2) ⋅ (360°) = 180°

b.

Complete the statement.

10.1.3 Guided Practice: Finding the Area of a Sector

One slice, or fraction of the circle, is bounded by two o diameters o radii

and an arc. Since each slice is the same

size, the central angle for each fractional piece of the circular pizza will

The images below show a stopwatch with different time increments marked. The blue shading on the stopwatches each forms a sector.

1.

o vary. o be equal.

c.

The pizza slices are an example of sectors. With your partner, write a definition of the term sector.

d.

Ask two other partner pairs to share their definition of a sector. Record a summary of their definition. You are the only person who should write in the first column of the table. Have the person initial next to your summary stating that your summary was correct.

My Summary of Their Definition of a Sector

Complete the table.

Sector Fraction of the Circle

Central Angle Measure of the Sector

Ratio of the Central Angle to the Whole Circle

Initials

Unit 10: Arcs and Angle Relationships in Circles

121

2.

Summarize what you notice about the relationship between the values in each row.

Given the radius of the stopwatch is 2 in., the area of the stopwatch is 4𝜋𝜋 in.2.

4.

3.

𝐴𝐴 = 𝜋𝜋𝑟𝑟 2 𝐴𝐴 = 𝜋𝜋(2)2 𝐴𝐴 = 4𝜋𝜋 in.2

Use the relationship between each sector and the whole stopwatch explored in problem 1 to find the area of each sector. Show your work.

Discuss with your partner how ratios could be used to find the area of the sectors, including any additional information that might be needed. Summarize your discussion.

122

Area of the Blue-Shaded Sector

Unit 10: Arcs and Angle Relationships in Circles

10.1.4 Your Turn!

1.

Five Slices

Kenya orders a medium pizza that has a 8 in. radius.

Complete the table on the number of slices she has.

Fraction of the Circle

Eight Slices

Area of the Sector

Central Angle

Three Slices

Seven Slices

One Slice

Unit 10: Arcs and Angle Relationships in Circles

123

10.1.5 Wrap-Up: Growth Mindset 1.

As you reflect on today’s lesson, circle sometimes, always, or never to indicate how often each statement describes you.

Statement When I work hard on a problem, I expect that the answer will be correct.

This Describes Me

Unit 10, Lesson 2: Arc Length Benchmark

MA.912.GR.6.4 Solve mathematical and real-world problems involving the arc length and area of a sector in a given circle.

Sometimes Always Never Learning Targets

There is no point in trying to do a problem if I know I won’t get the correct answer. Sometimes Always Never Even if I made a mistake, I could still learn from working Sometimes Always Never through a problem.

 

I can define arc length.

I can explain how the length of an arc is proportional to the ratio of the central angle to the whole circle, measured in degrees.

I can show that I understand how to solve a problem, even if there is an error in Sometimes Always Never my work.

If my answers are incorrect even after working very hard, I still choose to work hard on the next problem.

124

Sometimes Always Never

Unit 10: Arcs and Angle Relationships in Circles

10.2.2 Exploration: Arc Length and Proportional Relationships

10.2.1 Warm-Up: Notice and Wonder

The Chandelier Tree, located in Leggett California, is part of a drive-thru tree park. The tree is believed to have been carved in the early 1930’s by Charlie Underwood.

̅​̅​̅​̅​̅ is Circle 𝑃𝑃 with point 𝐾𝐾 on the circle and diameter 𝑌𝑌𝑌𝑌 shown.

1.

a.

What is 𝑚𝑚∠𝑌𝑌𝑌𝑌𝑌𝑌?

1.

What do you notice?

2.

What do you wonder?

b.

Determine the ratio of the measure of the central angle, ∠𝑌𝑌𝑌𝑌𝑌𝑌, to the whole circle. Simplify the ratio.

3.

Estimate the diameter of the tree at the base.

c.

An arc of a circle is the portion of the circumference between points. Point 𝐾𝐾 is a point on Circle 𝑃𝑃. Discuss with your partner which part of the circle is arc 𝑌𝑌𝑌𝑌𝑌𝑌.

Then, trace over this portion of the circle.

d.

What fraction of the circle’s circumference does arc 𝑌𝑌𝑌𝑌𝑌𝑌 represent?

4.

Explain why you chose that number as your estimate.

Unit 10: Arcs and Angle Relationships in Circles

125

⏜ .

Arc 𝑌𝑌𝑌𝑌𝑌𝑌 can also be notated as 𝑌𝑌𝑌𝑌𝑌𝑌 2. Circle 𝐿𝐿 is shown where points 𝑊𝑊 and 𝐶𝐶 are on the circle.

3.

4.

5.

On Circle 𝑅𝑅, the distance between each point measured along the circumference is 1 inch.

a.

What is the ratio of the central angle, ∠𝐶𝐶𝐶𝐶𝐶𝐶, to the whole circle? Simplify the ratio.

a.

b.

⏜ What fraction of the circle’s circumference does 𝑊𝑊𝑊𝑊 represent?

b.

Verify your answers to problems 1 and 2 with your partner and resolve any differences, if necessary.

With your partner, discuss any relationships you observe. Summarize your discussion.

126

c. d.

Mark a point anywhere along the circle’s circumference as point 𝑇𝑇. Then, starting at point 𝑇𝑇, mark an 8-inch section along the circle’s circumference. Label the other endpoint of the 8inch section as point 𝐺𝐺.

What fraction of the circle’s circumference does the ⏜ represent? length of 𝑇𝑇𝑇𝑇

Draw the central angle ∠𝑇𝑇𝑇𝑇𝑇𝑇.

What is the measure of the central angle ∠𝑇𝑇𝑇𝑇𝑇𝑇?

Unit 10: Arcs and Angle Relationships in Circles

e.

What is the ratio of the central angle to the whole circle?

10.2.3 Activity: Matching Central Angles and Ratios of Arc Length to Circumference 1.

f.

6.

Discuss with your partner the relationship between the length of the arc and the ratio of the central angle to the whole circle.

o arc length o diameter o radius

is the distance along the part of

a circle’s circumference or along any curve.

The measure of an arc is

a.

Work with your partner to match each circle card to the corresponding ratio indicated on the lettered card. Record the matches in the table.

Card Letter

Complete the statements.

The

Your teacher will give you two sets of cards for this activity. One set shows circles with a central angle. The other set of cards is a set of ratios of the arc length to the circumference.

o the inverse of o proportional to

ratio of the central angle to the whole circle.

the

Ratio of the Arc Length to the Circumference

Card Letter

A

F

B

G

C

H

D

I

E

J

b.

Ratio of the Arc Length to the Circumference

Summarize how you and your partner were able to match the circles to a corresponding ratio without being given the circumference.

Unit 10: Arcs and Angle Relationships in Circles

127

c.

Complete the statement.

10.2.4 Exploration: Arc Lengths With the Same Central Angle and Different Radii

The ratio of the central angle to the whole circle is the o inverse of o same as

the ratio of the length of an arc to

the circumference.

1.

Points 𝐷𝐷 and 𝑃𝑃 lie on Circle 𝑊𝑊. The radius ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 measures 5 inches (in.).

A smaller Circle 𝑊𝑊 with a radius measure of 2 in. rests inside the larger Circle 𝑊𝑊, where points 𝑀𝑀 and 𝐾𝐾 lie on the smaller circle.

The smaller Circle 𝑊𝑊 has central angle ∠𝐾𝐾𝐾𝐾𝐾𝐾 that measures 45° and the larger Circle 𝑊𝑊 has central angle ∠𝑃𝑃𝑃𝑃𝑃𝑃 that measures 45°.

Use the information to complete the following.

a.

128

⏜ and 𝐷𝐷𝐷𝐷 ⏜ are different lengths. Discuss with your 𝑀𝑀𝑀𝑀 partner whether you think the ratio of the arc length and circumference will be the same as or different than the ratio of the central angle of the whole circle.

Unit 10: Arcs and Angle Relationships in Circles

b.

Andrew and Sarah are discussing if the ratio of the arc length to the circumference is the same as the ratio of the central angle to the whole circle for both the smaller and larger circles.

Sarah says, “Yes. They will have the same ratios because they have the same central angle.”

10.2.5 Wrap-Up: Reflection 1.

Andrew says, “No. They do not have the same ratios because they are different circles with different radii.”

Explain who you agree with.

2. c.

Complete the table to support your reasoning in Part B. Ratio of Arc Length to Circumference

Choose the emoji that reflects your understanding of arc length.

Complete the statement.

I chose this emoji because

Ratio of Central Angle to Whole Circle

Circle 𝑊𝑊 with radius 2 in.

Circle 𝑊𝑊 with radius 5 in. d.

Explain whether your findings in Part C support your answers to Parts A and B. If not, explain why your answers to Parts A and B are incorrect.

Unit 10: Arcs and Angle Relationships in Circles

129

Unit 10 Lesson 3: Radian Measure

10.3.1 Warm-Up: Would You Rather?

Benchmark

MA.912.GR.6.4 Solve mathematical and real-world problems involving the arc length and area of a sector in a given circle.

1.

Would you Rather…

Deposit $3 in the Deposit $5 in the bank bank and have it and have it triple each OR quadruple each week for 4 weeks? week for 4 weeks?

Which would you rather choose? Justify your reasoning using mathematics.

Learning Targets  

I can define radian.

I can explain the relationship between the length of the arc intercepted by an angle to the radius.

130

Unit 10: Arcs and Angle Relationships in Circles

10.3.2 Exploration: Discovering Radian Measure

1.

Angles that are measured in degrees can also be measured in radians. Measuring in radians is similar to measuring distance using other units, such as inches or feet. Radians can be written as a decimal or in pi (𝜋𝜋) form.

With your partner, work through the Desmos activity “Discover Radian Measure.” Complete the steps to answer each question.

Radius (𝒓𝒓)

𝑪𝑪𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 (𝒄𝒄)

• •

Slider “𝑟𝑟” adjusts the radius of the circle. The radii for the activity range from 0 to 3, increasing by 0.5.

Slider “𝑐𝑐” adjusts the central angle.

The central angle, 𝐶𝐶𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 , is measured in radians, written as a decimal. Radians for the activity increase by 0.25.

Describe the relationship observed in the table between the measure of the arc length and the measure of the radius.

3.

Adjust slider 𝑟𝑟, radius, so that the arc length, 𝐴𝐴𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟ℎ , is twice the radius, 𝑟𝑟. Complete the table for three different measures of 𝑟𝑟.

Drag slider 𝑟𝑟 to the right and left to observe how the measurements of radius, central angle, and arc length change as the 𝑟𝑟 value changes. These values are in rows 2, 5, and 6 respectively of the activity.

Hit play on slider 𝑐𝑐 to watch how the measurements of radius, central angle, and arc length changes. These values are in rows 2, 5, and 6 respectively.

𝑨𝑨𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓

2.

Take a moment to look over the activity and the information provided.

•

Adjust slider 𝑐𝑐, 𝐶𝐶𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 , so that the arc length, 𝐴𝐴𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟ℎ , equals the radius, 𝑟𝑟. Complete the table for three different measures of 𝑟𝑟.

Radius (𝒓𝒓)

𝑪𝑪𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 (𝒄𝒄) 4.

𝑨𝑨𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓

Describe the relationship observed in the table between the measure of the arc length and the measure of the radius.

Unit 10: Arcs and Angle Relationships in Circles

131

5.

Complete the statements based on your work in questions 1-4.

A radian is a unit used to express the measure of an angle. Radians are based on the number of

o angles o radii

o diameter o circumference

of a circle being wrapped along the

10.3.3 Exploration: Discovering Radians in a Whole Circle 1.

Consider the diagram depicting three concentric circles each with center 𝐸𝐸. Point 𝐴𝐴 lies on the blue Circle 𝐸𝐸, and has radius 𝐸𝐸𝐸𝐸 = 1 unit. Point 𝐹𝐹 lies on the brown Circle 𝐸𝐸, and has radius 𝐸𝐸𝐸𝐸 = 2 units. Point 𝐻𝐻 lies on the green Circle 𝐸𝐸, and has radius 𝐸𝐸𝐸𝐸 = 3 units.

of the circle. The radian measure of a central angle is equal to the length of the

o arc o radius

intercepted by the central angle divided by the o radius. o arc length.

a.

6.

Describe how the diagram shows the relationship between the radius and arc length.

Complete the table with your partner.

Circle

Circumference

Radius 𝐸𝐸𝐸𝐸 = 1 unit

Radius 𝐸𝐸𝐸𝐸 = 2 units

132

Radius 𝐸𝐸𝐸𝐸 = 3 units Unit 10: Arcs and Angle Relationships in Circles

Ratio of Circumference to Radius

b.

Discuss any observations you made about the values in the table with your partner. Summarize your discussion.

10.3.4 Bring It Together: Relating Arc Length to the Radius 1.

c.

Write a conjecture about the ratio of the circumference to radius and a whole circle measured in radians.

Concentric circles with center 𝐸𝐸 are shown. Points 𝐴𝐴 and 𝑁𝑁 lay on the blue Circle 𝐸𝐸, and has radius 𝐸𝐸𝐸𝐸 = 1 unit.

Points 𝐹𝐹 and 𝑀𝑀 lay on the brown Circle 𝐸𝐸, and has radius 𝐸𝐸𝐸𝐸 = 2 units. Points 𝐻𝐻 and 𝐿𝐿 lay on the green Circle 𝐸𝐸, and has radius 𝐸𝐸𝐸𝐸 = 3 units.

Describe how the arclengths 𝐴𝐴𝐴𝐴, 𝐹𝐹𝐹𝐹 and 𝐻𝐻𝐻𝐻 are related.

Unit 10: Arcs and Angle Relationships in Circles

133

2.

Use words from the bank below to complete the statements. angle

arc

radian

radius

The length of the ____________ intercepted by a(n)

10.3.5 Wrap-Up: Cool Down 1.

Your friend was absent today and will need notes from class. Write a quick note to help them understand relationship between the length of an arc intercepted by an angle and the radius.

____________ is proportional to the ____________ of the circle. The ____________ measure of the angle is the constant of proportionality.

134

𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑 =

𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂 𝟐𝟐𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 = = 𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑 𝒓𝒓 𝟏𝟏

𝟑𝟑𝟑𝟑𝟑𝟑° = 𝟐𝟐𝟐𝟐 𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫 𝟏𝟏𝟏𝟏𝟏𝟏° = 𝝅𝝅 𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫

Unit 10: Arcs and Angle Relationships in Circles

Unit 10, Lesson 4: Solving Problems Involving Arc Length and Area of a Sector Benchmark

MA.912.GR.6.4 Solve mathematical and real-world problems involving the arc length and area of a sector in a given circle.

10.4.1 Warm-Up: Bell Work 1.

A small, family-owned restaurant sells a personal pan pizza with a radius of 10 centimeters for $5. A nearby restaurant cuts their pizza with a radius of 30 centimeters into 8 slices of equal size, and charges $5 per slice.

Which is a better deal? Explain your reasoning using mathematics.

Learning Target 

I can solve mathematical and real-world problems involving arc length and sector area.

Unit 10: Arcs and Angle Relationships in Circles

135

d.

10.4.2 Exploration: Finding a Formula for the Area of a Sector 1.

Summarize your discussion by writing a general formula to find the area of any sector.

Suppose a circle is cut into 8 equal slices to form sectors. a.

Complete the table.

Central Number of Fraction of Angle Slices the Circle in Degrees (°)

Write an Equation to Find the Area of the Sector

2 4 6

b.

c.

136

8

Area of a Sector

Compare the values in your table with your partner.

Discuss your equations in the last column and make any revisions if needed.

𝜃𝜃

𝐴𝐴 = 360 𝜋𝜋𝑟𝑟 2 , where 𝜃𝜃 is in degrees 𝐴𝐴 =

1

2

Discuss with your partner how the fractions in your equations relate to the central angles, and how the equations might have differed if the angle measure was given in radians.

Unit 10: Arcs and Angle Relationships in Circles

𝑟𝑟 2 𝜃𝜃, where 𝜃𝜃 is in radians

10.4.3 Guided Practice: Area of a Sector 1.

2.

3.

̅​̅​̅​̅ that Circle 𝑌𝑌 has radius 𝐶𝐶𝐶𝐶 measures 5 inches with central angle ∠𝑅𝑅𝑅𝑅𝑅𝑅, where 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 = 160°. Find the area of the shaded sector. Round your answer to the nearest thousandth.

10.4.4 Collaboration: Arc Length Work with your partner to complete the following.

Circle 𝑋𝑋 has a diameter of 12 centimeters. Points 𝐵𝐵 and 𝑃𝑃 are on the circle and form the sector 𝐵𝐵𝐵𝐵𝐵𝐵. The central angle ∠𝐵𝐵𝐵𝐵𝐵𝐵 in Circle 𝑋𝑋 measures 60°. Determine the area of the sector. Express your answer in terms of 𝜋𝜋.

The area of a sector with a radius of 8 inches is 74.84 square inches. Calculate the approximate measure of the central angle of the sector in degrees.

1.

Explain how the circumference relates to an arc length of a circle.

2.

Describe how to find the arc length of half a circle.

3.

Describe how to find the arc length of a quarter of a circle.

4.

Describe how to find the arc length of a circle with a central angle of 60°.

Unit 10: Arcs and Angle Relationships in Circles

137

5.

Write a general equation that can be used to find the arc length of a sector in degrees.

10.4.5 Guided Practice: Arc Length 1.

6.

Concentric circles with center 𝑃𝑃 are shown with 𝑚𝑚∠𝑃𝑃 = 54°. The smallest Circle 𝑃𝑃 has a radius ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃 that measures 15 meters (m). The largest Circle 𝑃𝑃 has ̅​̅​̅​̅ and the middle Circle 𝑃𝑃 has radius 𝑃𝑃𝑃𝑃 ̅​̅​̅​̅. Each radius 𝑃𝑃𝑃𝑃 circle has a radius that is 15 m longer than the immediate prior circle.

Discuss with your partner how your equation would change if the central angle had been given in radians.

Summarize your discussion by writing an equation to find the arc length if the angle measure is given in radians.

Calculate the arc length intercepted by ∠𝑃𝑃 on each circle.

Arc length of circle with radius ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃

Arc Length 𝜃𝜃

Arc Length = 𝐶𝐶 ⋅ 360°, where 𝜃𝜃 is in degrees Arc Length = 𝑟𝑟 ⋅ 𝜃𝜃, where 𝜃𝜃 is in radians

138

Arc length of circle ̅​̅​̅​̅ with radius 𝑃𝑃𝑃𝑃 Arc length of circle ̅​̅​̅​̅ with radius 𝑃𝑃𝑃𝑃

Unit 10: Arcs and Angle Relationships in Circles

2.

Points 𝑅𝑅 and 𝐶𝐶 are on Circle 𝑌𝑌. If 𝑚𝑚 ∠𝐶𝐶𝐶𝐶𝐶𝐶 is 160°, find ⏜. the length of 𝑅𝑅𝑅𝑅

10.4.6 Guided Practice: Solving Real-World Problems Involving Arc Length and Area of a Sector 1.

In Sarah’s family, the person celebrating their birthday always gets to cut the first piece of a circular birthday cake. Sarah is celebrating her birthday with both of her parents, her two brothers and her best friend. She cuts her piece of birthday cake and it formed a sector with a radius of 8 inches and a central angle with measure 85°.

If the 5 other party guests equally share the remaining portion of the cake, what is the approximate area of cake each person receives? Round your answer to two decimal places.

Unit 10: Arcs and Angle Relationships in Circles

139

2.

Dominic is riding a Ferris wheel at the county fair. The Ferris wheel forms a circle with center 𝐶𝐶. The Ferris wheel has a radius of 60 ft and rotates clockwise with seats that are equidistant from each other.

10.4.7 Wrap-Up: Reflection 1.

Fill in the table based on your experience with today’s lesson on arc length and area of sectors.

Today I deserve

for …

Today I

a.

After the ride begins from point 𝑨𝑨 and travels clockwise, it pauses at point 𝑩𝑩. How many feet has Dominic traveled when his seat pauses at point 𝑩𝑩? Write your answer in 𝜋𝜋 form and rounded to the nearest thousandth. 𝜋𝜋 or

b.

140

Today I was

feet

About how many feet has Dominic traveled if the full ride is 4 revolutions? 𝜋𝜋 or

by doing …

feet

by …

Unit 10: Arcs and Angle Relationships in Circles

Unit 10, Lesson 5: Solving Problems Involving Arc Measure and Central Angles

10.5.1 Warm-Up: Circles

Benchmark MA912.GR.6.2 Solve mathematical and real-world problems involving the measures of arcs and related angles.

1.

A square lattice with three circles is shown, where the circles pass through exactly 2, 3, or 4 points on the lattice, respectively.

Learning Target 

I can use the relationship between the measure of a central angle and its intercepted arc to solve problems.

a.

What do you notice about the circles?

b.

Connect the points on each circle to create a polygon. Describe any pattern you notice.

c.

What type of polygon would be inside a circle that passes through 5 points?

d.

On the lattice, draw a circle that passes through exactly 5 points.

Unit 10: Arcs and Angle Relationships in Circles

141

10.5.2 Exploration: Understanding Central Angles 1.

Circle 𝐵𝐵 is shown.

c.

Draw the radius ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵 from the center 𝐵𝐵 to point 𝑌𝑌. ̅​̅​̅​̅ from the center 𝐵𝐵 to Then, draw another radius 𝐵𝐵𝐵𝐵 point 𝐿𝐿.

d.

What is the measure, in degrees, of the central ⏜? angle ∠𝑌𝑌𝑌𝑌𝑌𝑌 in relation to 𝑌𝑌𝑌𝑌

e.

Next, along the circumference, label 0° as point 𝑈𝑈, 90° as point 𝑆𝑆, 250° as point 𝐸𝐸, and 340° as point 𝐻𝐻.

f.

B

a.

b.

142

Find the 130° mark and label it point 𝑌𝑌. Then, find the 220° mark and label it point 𝐿𝐿. Draw along the circumference from point 𝑌𝑌 to point 𝐿𝐿 going counterclockwise.

Exactly what fraction of the circle’s circumference is from point 𝑌𝑌 (130°) to point 𝐿𝐿 (220°)?

Draw two separate arcs along the circumference of the circle, one extending from point 𝑈𝑈 to point 𝑆𝑆 and the other extending from point 𝐸𝐸 to point 𝐻𝐻.

Then, draw the central angles that intersect with each part of the circumference you marked.

g.

What is the measure, in degrees, of each of the central angles, ∠𝑈𝑈𝑈𝑈𝑈𝑈 and ∠𝐸𝐸𝐸𝐸𝐸𝐸?

h.

Describe what you notice about all three central angles.

i.

Explain the relationship between a central angle and its intercepted arc.

Unit 10: Arcs and Angle Relationships in Circles

j.

Share your response with your partner. Listen to your partner’s explanation and write a summary of the explanation. Have your partner initial stating that your summary of their explanation is accurate.

Your Partner’s Explanation

k.

l.

2.

Circle 𝐶𝐶 is shown where points 𝑅𝑅, 𝐿𝐿, 𝑋𝑋, and 𝐻𝐻 are on the circle.

Partner’s Initials

Draw a diagram that shows the relationship between a central angle and its intercepted arc.

Discuss with your partner how arc measure relates to the relationship between a central angle and its intercepted arc. Summarize your discussion.

a.

⏜.

Name the angle that intercepts 𝑅𝑅𝑅𝑅

b.

Name the arc that is intercepted by ∠𝐻𝐻𝐻𝐻𝐻𝐻.

Unit 10: Arcs and Angle Relationships in Circles

143

c.

Brooklynn and Angelina were discussing how to name an arc that is intercepted by ∠𝑅𝑅𝑅𝑅𝑅𝑅. Each of their explanations are shown.

Brooklynn The arc that is intercepted by ⏜ because you just ∠𝑅𝑅𝑅𝑅𝑅𝑅 is 𝑅𝑅𝑅𝑅 use the first and last letters of the angle to name the arc.

Angelina An arc that is intercepted ⏜ or by ∠𝑅𝑅𝑅𝑅𝑅𝑅 could be 𝑅𝑅𝑅𝑅𝑅𝑅 ⏜ 𝑅𝑅𝑅𝑅𝑅𝑅 because the angle intercepts two arcs.

10.5.3 Bring It Together: Arc Measure and Central Angles

An arc of a circle is a section of the circumference of the circle.

The length of an arc is simply the length of the section of the circumference.

Discuss with your partner the conversation between Brooklynn and Angelina. Summarize your discussion.

In a circle, the measure of an arc (in degrees or radians) is equal to the measure of the central angle that intercepts the arc.

In a circle, any arc whose corresponding central angle has a measure greater than 180° is considered a major arc and should be named using three letters.

In a circle, any arc whose corresponding central angle has a measure less than 180° is considered a minor arc and should be named using two letters.

144

Unit 10: Arcs and Angle Relationships in Circles

1.

Circle 𝐺𝐺 has points 𝑃𝑃, 𝐵𝐵, and 𝑈𝑈 on the circle with 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 = 120° and ⏜ = 155°. Ryan and 𝑚𝑚 𝐵𝐵𝐵𝐵 Katelyn are finding the ⏜ , but measure of 𝑈𝑈𝑈𝑈 they found different answers.

10.5.4 Guided Practice: Finding Angle Measures

Ryan

Katelyn

155 + 120 = 275 275 − 180 = 95 ⏜ = 95° 𝑚𝑚 𝑈𝑈𝑈𝑈

155 + 120 = 275 360 − 275 = 85 ⏜ = 85° 𝑚𝑚 𝑈𝑈𝑈𝑈

1.

Circle 𝐷𝐷 has diameters ̅​̅​̅​̅​̅ 𝐹𝐹𝐹𝐹 and ̅​̅​̅​̅ 𝐺𝐺𝐺𝐺 that both intersect at ⏜ ⏜.

Point 𝐷𝐷. Given the 𝑚𝑚 𝑀𝑀𝑀𝑀 = 110°, find the 𝑚𝑚 𝐻𝐻𝐻𝐻

2.

Circle 𝑇𝑇 has points 𝐴𝐴, 𝑉𝑉, and 𝐶𝐶 on the circle with ⏜ = 132.3° and 𝑉𝑉𝑉𝑉 ⏜ does not 𝑚𝑚∠𝑉𝑉𝑉𝑉𝑉𝑉 = 21.7°. If 𝑚𝑚 𝑉𝑉𝑉𝑉 ⏜ , what is 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶? intersect 𝑉𝑉𝑉𝑉

Do you agree with Ryan or Katelyn? Justify your answer.

Unit 10: Arcs and Angle Relationships in Circles

145

3.

4.

Circle 𝐶𝐶 has radii ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 and ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶. Find the value of 𝑥𝑥 if ⏜ 𝑚𝑚∠𝐿𝐿𝐿𝐿𝐿𝐿 = 114° and 𝑚𝑚 𝐿𝐿𝐿𝐿 = (6𝑥𝑥 + 6)°.

10.5.5 Your Turn!

1.

In Circle 𝑀𝑀, 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 = 100° and 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 = 40°.

Calculate 𝑚𝑚 ⏜ 𝑅𝑅𝑅𝑅𝑅𝑅 given that 𝑀𝑀 is the center and ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 and ̅​̅​̅​̅ are diameters. 𝐴𝐴𝐴𝐴

Isabella is creating a folding fan for art class. She needs to determine the measure of the arc of the fan before she can cut the paper.

She uses the expression (8𝑥𝑥 + 102)° as the measure of ⏜ , she uses the the central angle, ∠𝑁𝑁𝑁𝑁𝑁𝑁. For the 𝑚𝑚 𝐺𝐺𝐺𝐺 expression (32𝑥𝑥 + 54)°. ⏜?

What is the degree measure of the arc of the fan, 𝐺𝐺𝐺𝐺

146

Unit 10: Arcs and Angle Relationships in Circles

2.

Spirit is tied to a post at point 𝑉𝑉 and is grazing in a circular pattern. He decides to walk from his position at point 𝑆𝑆 to the tree at point 𝑇𝑇 along the circumference of the circular area he can roam. ⏜ = (9𝑥𝑥 − 23)°, find the If the 𝑚𝑚∠𝑆𝑆𝑆𝑆𝑆𝑆 = (5𝑥𝑥 + 33)° and 𝑚𝑚 𝑆𝑆𝑆𝑆 ⏜. measure of the arc that Spirit walked, 𝑚𝑚 𝑆𝑆𝑆𝑆

10.5.6 Wrap-Up: Ticket Out the Door 1.

Circle 𝑋𝑋 has points 𝑅𝑅, 𝑃𝑃, and 𝐴𝐴 on the circle. The ⏜ = 141°, measure of the central angle ∠𝐴𝐴𝐴𝐴𝐴𝐴 = 50°, 𝑚𝑚 𝑅𝑅𝑅𝑅 ⏜ does not intersect 𝐴𝐴𝐴𝐴 ⏜ . Find 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅. and 𝑅𝑅𝑅𝑅

Unit 10: Arcs and Angle Relationships in Circles

147

Unit 10, Lesson 6: Solving Problems Involving Inscribed Angles

10.6.1 Warm-Up: Angles on a Clock 1.

Use the clock to answer the following questions.

Benchmark MA.912.GR.6.2 Solve mathematical and real-world problems involving the measures of arcs and related angles.

Learning Target 

I can use the relationship between the measure of an inscribed angle and its intercepted arc to solve problems.

148

a.

If the time is 10: 10 and the hour hand does not move, what is the exact angle between the minute hand and hour hand?

b.

If the hour hand does move, what is the exact angle if the time is 7: 30?

c.

If the hour hand does move, what is the exact angle if the time is 7: 35?

Unit 10: Arcs and Angle Relationships in Circles

10.6.2 Exploration: Inscribed Angles 1.

c.

Write a conclusion about the measures of the central angle, the inscribed angle, and the arc measure.

d.

Share your response to Part C with two classmates.

Listen to their conclusion and summarize it in the table. You are the only person who should write in the first column of the table. Have the person initial next to your summary stating your summary was correct.

Circle 𝐿𝐿 is shown where 𝑃𝑃, 𝐶𝐶, and 𝑀𝑀 are points on the circle.

A chord is a line segment on the interior of a circle with both endpoints lying on the circle.

An inscribed angle in a circle is an angle which is formed in the interior of a circle when two chords share an endpoint.

Central Angle 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 = 110°

My Summary of Their Conclusion

Initials

Inscribed Angle 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 = 55° ⏜ = 110° Arc Measure 𝑃𝑃𝑃𝑃 a.

b.

What is the connection between the central angle and the arc measure?

Describe the relationship between the central angle and the inscribed angle.

e. Using Circle 𝐿𝐿, complete the table.

Central Angle Inscribed Angle Arc Measure 216°

81° 145°

Unit 10: Arcs and Angle Relationships in Circles

143°

149

2.

∠𝐴𝐴𝐴𝐴𝐴𝐴 is inscribed in semicircle 𝐷𝐷, as shown.

a.

Describe the angle inscribed in the semicircle.

b.

If the entire Circle 𝐷𝐷 is shown, what is the measure of ⏜? 𝐴𝐴𝐴𝐴

c.

Discuss with your partner the relationship between an inscribed angle and a semicircle.

10.6.3 Guided Practice: Inscribed Angles 1.

Circle 𝐴𝐴 is shown where points 𝑅𝑅, 𝑊𝑊, and 𝐻𝐻 are on the circle. ∠𝑅𝑅𝑅𝑅𝑅𝑅 is inscribed inside Circle 𝐴𝐴 ⏜ = 80°. with 𝑚𝑚 𝑅𝑅𝑅𝑅 Find the value of 𝑥𝑥.

2.

Points 𝑌𝑌, 𝐺𝐺, 𝐶𝐶, and 𝑊𝑊 lie on Circle 𝐻𝐻. ∠𝐺𝐺𝐺𝐺𝐺𝐺 is inscribed in the circle.

Find 𝑥𝑥, given 𝑚𝑚 ⏜ 𝐺𝐺𝐺𝐺𝐺𝐺 = 192° and 𝑚𝑚∠𝐺𝐺𝐺𝐺𝐺𝐺 = (31𝑥𝑥 + 3)°.

Inscribed Angle Theorem The measure of an inscribed angle is one-half the measure of the intercepted arc.

150

Unit 10: Arcs and Angle Relationships in Circles

3.

Boats use angle measures and arcs when sailing in treacherous waters. When a boat captain turns on the boat light, it forms inscribed angle ∠𝑁𝑁𝑁𝑁𝑁𝑁. Use the information for Circle 𝐻𝐻 to find 𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁.

4.

Craig and Valentina are reviewing circles to identify inscribed angles.

Circle A

Circle B

Circle C

Circle D

(13𝑥𝑥 − 10)°

Valentina identified Circle C is an inscribed angle. Craig identified both Circle A and Circle C as inscribed angles.

(7𝑥𝑥 − 10)°

Do you agree with Valentina, Craig, or neither? Justify your reasoning.

Unit 10: Arcs and Angle Relationships in Circles

151

10.6.4 Your Turn!

1.

Circle 𝑇𝑇 is shown where ∠𝐸𝐸𝐸𝐸𝐸𝐸 is a central angle, ∠𝐸𝐸𝐸𝐸𝐸𝐸 is inscribed in the circle, and 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 = 62°. Find 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸.

3.

Equilateral triangle 𝐶𝐶𝐶𝐶𝐶𝐶 is inscribed in Circle 𝐴𝐴. Circle 𝐴𝐴 consists of three equal inscribed angles.

Determine the value of the inscribed angle.

2.

Circle 𝐴𝐴 has points 𝐷𝐷, 𝑆𝑆, and 𝐿𝐿 on the circle. ∠𝐷𝐷𝐷𝐷𝐷𝐷 is inscribed in Circle 𝑀𝑀 measuring 95°. Given ⏜ = 75°, find 𝑚𝑚 𝐷𝐷𝐷𝐷 ⏜. 𝑚𝑚 𝑆𝑆𝑆𝑆

152

Unit 10: Arcs and Angle Relationships in Circles

10.6.5 Wrap-Up: Reflection 1.

What was the muddiest (most difficult or confusing) point in today’s lesson on solving problems using the relationship between the measure of an inscribed angle and its intercepted arc?

Unit 10, Lesson 7: Solving Problems Involving Intersecting Chords Benchmark

MA.912.GR.6.2 Solve mathematical and real-world problems involving the measures of arcs and related angles.

Learning Target 

I can solve problems involving chords of a circle and their related angles.

Unit 10: Arcs and Angle Relationships in Circles

153

10.7.1 Warm-Up: Sectors 1.

10.7.2 Exploration: Chords

What do you notice about the circle graph?

Music 1 hour

Electives 2 hours

Lunch 1 hour

Circle 𝑊𝑊 is shown where points 𝑅𝑅, 𝐺𝐺, 𝐾𝐾, and 𝐻𝐻 lie on the circle.

Core Subjects 4 hours

2.

3.

Identify a sector that is incorrectly sized.

1.

̅​̅​̅​̅ and ̅​̅​̅​̅ What similarities do you notice between 𝑅𝑅𝑅𝑅 𝐺𝐺𝐺𝐺 ?

2.

What differences do you notice between ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 and ̅​̅​̅​̅ 𝐺𝐺𝐺𝐺 ?

Redraw the circle graph so that the sizes are proportional to the number of hours.

A chord is a line segment that connects two points on the curve of a circle.

154

Unit 10: Arcs and Angle Relationships in Circles

3.

Complete the statements.

̅​̅​̅​̅ is 𝑅𝑅𝑅𝑅

o a diameter . o a chord o both a diameter and a chord

̅​̅​̅​̅ 𝐺𝐺𝐺𝐺 is

o a diameter o a chord o both a diameter and a chord

10.7.3 Collaboration: Angles in Intersecting Chords of the circle.

1.

With your partner, complete the following using Circle D. a.

of the circle.

Draw and label two chords that intersect inside the circle, making sure the endpoints fall on the points.

When drawing the chords, do not draw the same chords as your partner and avoid drawing diameters.

Unit 10: Arcs and Angle Relationships in Circles

155

b.

Choose one of the angles to measure and label.

Then, complete the table.

Measure of Angle Arc it Measure Intercepts

e.

Angles of Intersecting Chords Theorem

Measure of the Arc the Vertical Angle Intercepts

The angle formed inside of a circle by two intersecting chords is o double o equal o half

Your Angle

d.

156

the sum of

the measures of the arcs intercepted by the

angle and its vertical angle.

Your Partner’s Angle

c.

Complete the statement.

Discuss the following with your partner.

•

What do you notice about the values in the table?

•

What do you think the relationship is between the angle measure and the two intercepted arcs?

Write a formula that generalizes your discovery from the discussion you had in Part C.

Unit 10: Arcs and Angle Relationships in Circles

10.7.4 Guided Practice: Solving Problems Involving Angles in Intersecting Chords 1.

̅​̅​̅​̅ intersect at Circle 𝑆𝑆 is shown, where chords ̅​̅​̅​̅ 𝐴𝐴𝐴𝐴 and 𝑌𝑌𝑌𝑌 ⏜ = 71° point 𝑃𝑃. Find the value of 𝑥𝑥 for ∠𝐿𝐿𝐿𝐿𝐿𝐿, given 𝑚𝑚 𝐴𝐴𝐴𝐴 ⏜ = 129°. and 𝑚𝑚 𝐿𝐿𝐿𝐿

2.

A tree is cut down and a circular tree stump is left. The top of the tree stump has two cracks that represent two chords, ̅​̅​̅​̅ 𝐹𝐹𝐹𝐹 and ̅​̅​̅​̅​̅ 𝐷𝐷𝐷𝐷, intersecting at point 𝑋𝑋 in Circle 𝐶𝐶. If ⏜ ⏜ = 39°, what is 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹? 𝑚𝑚 𝐹𝐹𝐹𝐹𝐹𝐹 = 175° and 𝑚𝑚 𝐾𝐾𝐾𝐾

Unit 10: Arcs and Angle Relationships in Circles

157

3.

Circle 𝐴𝐴 is shown, where chords ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 and ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 intersect at ⏜ point 𝐺𝐺. Find the value of 𝑥𝑥, given 𝑚𝑚 𝑆𝑆𝑆𝑆 = 70°, ⏜ = 136°, and 𝑚𝑚∠𝑀𝑀𝑀𝑀𝑀𝑀 = (7𝑥𝑥 + 19)°. 𝑚𝑚 𝑀𝑀𝑀𝑀

158

10.7.5 Your Turn!

1.

Circle 𝑅𝑅 is shown, where chords ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 and ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 intersect at ⏜ point 𝐺𝐺. Find the value of 𝑥𝑥, given 𝑚𝑚 𝑇𝑇𝑇𝑇 = 80°, ⏜ = (17𝑥𝑥 + 10)°, and 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 130°. 𝑚𝑚 𝐶𝐶𝐶𝐶

Unit 10: Arcs and Angle Relationships in Circles

10.7.6 Wrap-Up: Reflection Draw an X on each line to indicate where your current learning is in relation to solving problems involving chords.

1. I can define a chord of a circle.

I need help.

I can’t get started.

I’m getting there, but I need more practice.

I understand this, and I feel confident.

I’m getting there, but I need more practice.

I understand this, and I feel confident.

Benchmark

MA.912.GR.6.2 Solve mathematical and real-world problems involving the measures of arcs and related angles.

Learning Target

2. I can define the relationship of angles and arcs formed by the intersection of two chords.

I need help.

I can’t get started.

Unit 10, Lesson 8: Solving Problems Involving Intersection of Secants



I can solve problems involving secants drawn from a point external to the circle and their related angles.

3. I can solve problems involving chords of a circle.

I need help.

I can’t get started.

I’m getting there, but I need more practice.

I understand this, and I feel confident.

Unit 10: Arcs and Angle Relationships in Circles

159

10.8.1 Warm-Up: Satellite Operations Engineer 1.

Veronica Navarro is a satellite operations engineer.

Before beginning her career, she received a bachelor’s degree. One piece of advice she would give to high school students is to take as many classes in high school as possible because they are free. In her opinion, a degree beyond high school is needed in many careers because employers want to know a person can be trained and is capable of learning.

Veronica’s employer paid for her to receive additional degrees as a means to further develop her abilities and deepen her learning.

10.8.2 Exploration: Angles Formed by Secants A secant is a line that passes through two points on a circle. In the interior of the circle, a secant line forms a chord.

1.

∠𝑀𝑀𝑀𝑀𝑀𝑀 intersects Circle 𝐾𝐾, creating two intersecting arcs, where points 𝑀𝑀, 𝑍𝑍, 𝑌𝑌, and 𝑆𝑆 are on the circle. The 𝑚𝑚∠𝑍𝑍𝑍𝑍𝑍𝑍 = 38°, 𝑚𝑚∠𝑆𝑆𝑆𝑆𝑆𝑆 = 80°, and 𝑚𝑚∠𝑀𝑀𝑀𝑀𝑀𝑀 = 21°.

Why do you think it is important for companies to know you are capable of being trained?

a.

Complete the table.

Arc

Central Angle Measure

Arc Measure

⏜ 𝑀𝑀𝑀𝑀 ⏜ 𝑍𝑍𝑍𝑍

b.

Add and subtract the two arc measures.

⏜ +⏜ 𝑀𝑀𝑀𝑀 𝑍𝑍𝑍𝑍= ______ 160

⏜ − 𝑍𝑍𝑍𝑍 ⏜ = ______ 𝑀𝑀𝑀𝑀

Unit 10: Arcs and Angle Relationships in Circles

c.

d.

Do the operations in Part B compare to the measure of ∠𝑀𝑀𝑀𝑀𝑀𝑀? If so, explain.

10.8.3 Guided Practice: Solving Problems Involving Angles Formed by Secants 1.

Circle 𝐴𝐴 has points 𝐶𝐶, 𝐸𝐸, 𝐹𝐹, and 𝐵𝐵 on the circle. Secants ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 and ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵 intersect at point 𝐷𝐷 outside the circle. ⏜ = 125°, 𝑚𝑚 𝐸𝐸𝐸𝐸 ⏜ = 27°, and 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑥𝑥°. What is the 𝑚𝑚 𝐶𝐶𝐶𝐶 value of 𝑥𝑥?

With your partner, write a formula that can be used to find the measure of an exterior angle of two secant lines.

Exterior Secant Angle Theorem The measure of an angle formed by two secants intersecting in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.

Unit 10: Arcs and Angle Relationships in Circles

161

2.

A satellite, point 𝑊𝑊, sends signals ̅​̅​̅​̅​̅ 𝑉𝑉𝑉𝑉 and ̅​̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 to Earth. ̅​̅​̅​̅​̅ hits tower 𝐸𝐸 before reaching tower 𝑉𝑉.

Signal 𝑉𝑉𝑉𝑉 Similarly, signal ̅​̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 hits tower 𝐺𝐺 before reaching tower 𝑆𝑆. Scientists are interested in finding the measure of the intersecting arcs created by the two signals. They ⏜ = (12𝑥𝑥 + 7)°, and 𝑚𝑚 𝐸𝐸𝐸𝐸 ⏜ = (3𝑥𝑥 + 1)°. know 𝑚𝑚∠𝑊𝑊 = 30°, 𝑚𝑚 𝑉𝑉𝑉𝑉 ⏜ and 𝑚𝑚 𝐸𝐸𝐸𝐸 ⏜.

Find 𝑚𝑚 𝑉𝑉𝑉𝑉

(12𝑥𝑥 + 7)°

162

(3𝑥𝑥 + 1)°

10.8.4 Your Turn!

1.

Circle 𝑇𝑇 has points 𝐶𝐶, 𝐾𝐾, 𝑁𝑁, and 𝐸𝐸 on the circle. Secants ̅​̅​̅​̅ and ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸 intersect at point 𝑅𝑅 outside the circle. The ⏜ = 92°, 𝑚𝑚 𝐾𝐾𝐾𝐾 ⏜ = 22°, and 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑥𝑥°. What is the 𝑚𝑚 𝐶𝐶𝐶𝐶 value of 𝑥𝑥?

͵Ͳ°

Unit 10: Arcs and Angle Relationships in Circles

2.

Circle 𝐻𝐻 has points 𝑍𝑍, 𝐾𝐾, 𝑉𝑉, and 𝑅𝑅 on the circle. Secants ̅​̅​̅​̅ and 𝑅𝑅𝑅𝑅 ̅​̅​̅​̅ intersect at Point 𝐴𝐴 outside the circle. The 𝑍𝑍𝑍𝑍 ⏜ = (5𝑥𝑥 + 10)°, 𝑚𝑚 𝐾𝐾𝐾𝐾 ⏜ = (3𝑥𝑥 + 4)°, and 𝑚𝑚∠𝑍𝑍𝑍𝑍𝑍𝑍 = 30°. 𝑚𝑚 𝑍𝑍𝑍𝑍 ⏜.

Find the measure of 𝑍𝑍𝑍𝑍

10.8.5 Wrap-Up: Cool Down 1.

Write a short note to a friend who missed class summarizing today’s lesson.

Unit 10: Arcs and Angle Relationships in Circles

163

Unit 10 Lesson 9: Solving Problems Involving Tangents

Benchmark MA.912.GR.6.2 Solve mathematical and real-world problems involving the measures of arcs and related angles.

1.

In the circles, the outer numbers are used to calculate the number in the middle. Find the rule to determine what number should replace the question mark in the third circle.

͹

͸

ͳ

Learning Target 

10.9.1 Warm-Up: Number Puzzle

ͷ

ʹ

I can solve problems related to tangents and secants and the measures of their intercepted arcs.

164

Unit 10: Arcs and Angle Relationships in Circles

ͳʹ ͸

͵͸

ͺ ͵

ͻ

Ͷ

ǫ

ͺ ͳ

10.9.2 Exploration: Intersecting Tangent Lines

1.

With your partner, answer the following.

a.

Without measuring, which dashed line represents the shortest distance from point 𝑃𝑃 to ̅​̅​̅​̅ 𝐴𝐴𝐴𝐴? Explain your answer.

b.

Point 𝑃𝑃 is the center of a circle that intersects ̅​̅​̅​̅ 𝐴𝐴𝐴𝐴.

In the table, describe your observations of each dashed segment in relation to the center of Circle 𝑃𝑃.

Segment ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃

Relationship with the Center of Circle 𝑷𝑷

̅​̅​̅​̅​̅ 𝑃𝑃𝑃𝑃 ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃 ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃 Unit 10: Arcs and Angle Relationships in Circles

165

c.

Discuss with your partner the relationship the blue segment has with the center of Circle 𝑃𝑃.

10.9.3 Collaboration: Angles Formed by Tangents and Secants 1.

Circle 𝐺𝐺 with tangents ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 and ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 is shown. Use the diagram to answer the following.

A tangent is a line or segment that intersects a circle at exactly one point.

If a line is tangent to a circle, then it is perpendicular to a radius, which ends at the point of tangency.

a.

Complete the table.

Measure Major Arc 𝑯𝑯𝑯𝑯𝑯𝑯 Minor Arc 𝑯𝑯𝑯𝑯 ∠𝑯𝑯𝑯𝑯𝑯𝑯

166

Unit 10: Arcs and Angle Relationships in Circles

b.

Use the values for the major and minor arcs of Circle 𝐺𝐺 to determine a relationship between the arcs and the measure of the exterior angle formed by the two tangents. The relationship can be described using words or equations.

c.

Ask two other partner pairs to share their relationships. Record their relationships and ask them to initial verifying you have written it correctly. If their description or equation is different than yours, discuss the differences and determine if your description or equation needs to be revised.

Description or Equation

d.

2.

Circle 𝐴𝐴 shows ∠𝐺𝐺𝐺𝐺𝐺𝐺 formed by the intersection of ̅​̅​̅​̅ and secant ̅​̅​̅​̅ tangent 𝑆𝑆𝑆𝑆 𝐺𝐺𝐺𝐺 outside the circle at point 𝑆𝑆. ⏜ = 164°, 𝑚𝑚 𝑇𝑇𝑇𝑇 ⏜ = 74°, and 𝑚𝑚∠𝐺𝐺𝐺𝐺𝐺𝐺 = 45°. 𝑚𝑚 𝐺𝐺𝐺𝐺

a.

⏜ and 𝑇𝑇𝑇𝑇 ⏜ of Circle 𝐴𝐴 to Use the values for 𝐺𝐺𝐺𝐺 determine the relationship between the arcs and the measure of the exterior angle formed by the tangent and the secant.

b.

Discuss the relationship with your partner.

Initials

Explain how to find the measure of an angle formed by two tangents drawn to a circle from a point outside that circle. Be sure to use the term(s) “intercepted arc(s)” at least once in the description.

The formula for the measure of an External Angle to a Circle formed by two tangents, two secants, or a tangent and a secant is

1

Measure of angle formed = 2 ሺdifference of intercepted arcsሻ

OR 1

Measure of angle formed = 2 ሺmajor arc − minor arcሻ

Unit 10: Arcs and Angle Relationships in Circles

167

2.

10.9.4 Your Turn!

1.

Circle 𝐹𝐹 has points 𝐶𝐶 and 𝑌𝑌 on the circle. Tangents ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 ̅​̅​̅​̅ and 𝑌𝑌𝑌𝑌 intersect at point 𝐻𝐻 outside the circle. ⏜ = 239.21°, 𝑚𝑚 𝐶𝐶𝐶𝐶 ⏜ = 𝑥𝑥°, and 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 59.21°. Find the 𝑚𝑚 𝐶𝐶𝐶𝐶𝐶𝐶 value of 𝑥𝑥.

Circle 𝑊𝑊 has points 𝐷𝐷, 𝑉𝑉, and 𝐵𝐵 on the circle. Tangent ̅​̅​̅​̅ ̅​̅​̅​̅ intersect at Point 𝑍𝑍 outside the circle. 𝐵𝐵𝐵𝐵 and secant 𝐷𝐷𝐷𝐷 ⏜ = 148°, 𝑚𝑚 𝑉𝑉𝑉𝑉 ⏜ = 62°, and 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = 𝑥𝑥°. What is 𝑚𝑚 𝐷𝐷𝐷𝐷 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷?

168

Unit 10: Arcs and Angle Relationships in Circles

10.9.5 Wrap-Up: Reflection

Think back over this unit. If necessary, look back over the different lessons. Then, complete the sentences by checking all that apply.

1.

2.

I don’t understand

I can explain

 arc length  area of a sector  converting between radians and degrees  measures of arcs and related angles

Unit 10, Lesson H1: Constructing Tangent Lines Benchmark

...YET.

 how to solve problems involving arc length.  how to solve problems involving: area of a sector.  converting angle measures between degrees and radians.  how to solve problems involving measures of arcs and related angles including central, inscribed, and intersections of a chords, secants, and tangents.

MA.912.GR.5.5 Given a point outside a circle, construct a line tangent to the circle that passes through the given point.

Learning Target 

I can construct a line tangent to a circle that passes through a given point outside the circle.

Unit 10: Arcs and Angle Relationships in Circles

169

10.H1.1 Warm-Up: Radio Signals 1.

Two-way radio communications help climbers stay in touch with base camp and contact the other expeditioners climbing Mt. Everest. The separate expeditions reserve a central channel where the various groups can share information on weather conditions and route changes or call for rescue.

10.H1.2 Collaboration: Tangent Lines of Circles 1.

The equation shows the maximum theoretical range of a radio signal, in miles, using a line-of-sight calculation, which considers the curvature of the Earth and presumes the earth to be a perfect sphere – which of course it is not. This also assumes the radio signal is transmitted with sufficient power that maintains enough energy to travel the distance. 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻(𝑚𝑚𝑚𝑚) = 1.23 × √ℎ𝑒𝑒𝑒𝑒𝑒𝑒ℎ𝑡𝑡(𝑓𝑓𝑓𝑓)

a.

Use the formula to find the distance the radio signal would go based on your height.

b.

Explain the relationship of the radio signal range (red line) and Earth.

⃡ Circle 𝐵𝐵 is shown, with point 𝐾𝐾 on the circle and 𝑅𝑅𝑅𝑅 tangent to circle 𝐵𝐵.

a.

Write two facts about circle 𝐵𝐵 and tangent line ⃡𝑅𝑅𝑅𝑅 .

1.

2.

b.

2.

Share your facts with your partner and write down one additional fact.

3.

Complete the statement.

A line tangent to a circle touches a circle at point(s), and the line is o parallel o perpendicular radius of the circle.

170

Unit 10: Arcs and Angle Relationships in Circles

o one o two o three to the

3.

To construct a line tangent to a circle, you need a compass and a ruler. a.

Work with your partner and use the steps below to construct a line that is tangent to circle 𝑆𝑆 and goes through point 𝐴𝐴.

4.

Jamie wrote the steps for constructing a line tangent to a circle on note cards, including the steps of the perpendicular bisector, but the cards got out of order. a.

Place the compass on the midpoint of the line segment and adjust its lengths to reach the center.

Place the compass at one end of the line segment and adjust the compass to slightly longer than half the line segment.

Place the ruler where the arcs are above and below the line cross, and draw the perpendicular line.

Keeping the same width and compass point at the midpoint, draw a circle.

Draw arcs above and below the line.

Use the ruler to draw a line from the center, point 𝑆𝑆, 1. to the given point.

2.

3.

4.

Where the new circle crosses the original circle, create a point and draw the tangent line.

Draw a line connecting the center of the circle and the point outside the circle.

Keeping the same compass width, draw arcs above and below the line from the other end of the line segment.

Construct the perpendicular bisector of ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 to find its midpoint.

Set the compass width to the distance from the midpoint of ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 to the circle’s center, point 𝑆𝑆.

Keeping the compass at the midpoint, mark an arc across the circle at two places, creating 𝑀𝑀 and 𝑉𝑉.

Work with your partner to order the steps 1 − 8.

b.

Explain how Jamie’s method is different.

c.

Explain why Jamie’s method works.

5. Draw the two tangent lines from 𝐴𝐴 to 𝑀𝑀 and 𝐴𝐴 to 𝑉𝑉.

Unit 10: Arcs and Angle Relationships in Circles

171

2.

10.H1.3 Your Turn!

1.

Given circle 𝑊𝑊 and point 𝐶𝐶 outside the circle, construct a line tangent to circle 𝑊𝑊 going through point using Jamie’s method.

172

Naomi and Xavier were practicing constructing a line tangent to circle 𝐵𝐵 going through point 𝐴𝐴, but their results did not match.

Naomi’s Construction

Xavier’s Construction

a.

Who correctly drew a line tangent to circle 𝐵𝐵 going through point 𝐴𝐴?

b.

Explain the mistake the other student made.

Unit 10: Arcs and Angle Relationships in Circles

10.H1.4 Wrap-Up: Ticket Out the Door 1.

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

Given circle 𝐺𝐺 and point 𝑅𝑅 outside the circle, construct a line tangent to circle 𝐺𝐺 going through point 𝑅𝑅.

Unit 10: Arcs and Angle Relationships in Circles

173

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

174

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

Unit 10: Arcs and Angle Relationships in Circles

Unit 11 - Justifying Relationships in Polygons With Circles Lesson 11.1 Constructing a Circumscribed Circle of a Triangle 11.2 Constructing the Incenter of a Triangle 11.H1 Inscribing Polygons in Circles 11.3 Solving Problems Involving Triangles Inscribed in Circles 11.4 Solving Problems Involving Quadrilaterals Inscribed in Circles 11.5 Interior and Exterior Angles of Triangles 11.6 Triangle Inequality Theorem 11.7 Centroid of a Triangle 11.8 Triangle Midsegment Theorem 11.9 Solving Problems Using Triangle Relationships

Unit 11: Justifying Relationships in Polygons With Circles

Page Number 177 182 187 191 197 202 207 213 220 225

175

The following B.E.S.T. Standards will be covered in this unit: MA.912.GR.5.3 Construct the inscribed and circumscribed circles of a triangle.

• Clarification 1: Instruction includes using a compass and straightedge, string, reflective devices, paper folding or dynamic geometric software.

MA.912.GR.6.3 Solve mathematical problems involving triangles and quadrilaterals inscribed in a circle.

• Clarification 1: Instruction includes triangles in a circle and semicircle.

MA.912.GR.1.3 Prove relationships and theorems about triangles. Solve mathematical and real-world problems involving postulates, relationships and theorems of triangles.

• Clarification 1: Postulates, relationships and theorems include measures of interior angles of a triangle sum to 180°; measures of a set of exterior angles of a triangle sum to 360°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

• Clarification 2: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs.

• Clarification 3: Instruction focuses on helping a student choose a method they can use reliably.

MA.912.LT.4.10 Judge the validity of arguments and give counterexamples to disprove statements.

• Clarification 1: Within the Geometry course, instruction focuses on the connection to proofs within the course.

MA.912.GR.5.4 (Honors only) Construct a regular polygon inscribed in a circle. Regular polygons are limited to triangles, quadrilaterals, and hexagons.

• Clarification 1: When given a circle, the center must be provided.

• Clarification 2: Instruction includes using a compass and straightedge, string, reflective devices, paper folding or dynamic geometric software.

176

Unit 11: Justifying Relationships in Polygons With Circles

Unit 11, Lesson 1: Constructing a Circumscribed Circle of a Triangle Benchmark MA.912.GR.5.3 Construct the inscribed and circumscribed circles of a triangle.

11.1.1 Warm-Up: Lake Okeechobee and Deep Lake Lake Okeechobee means "big water" in the language of the Seminole tribe. Lake Okeechobee has a surface area of 730 square miles. Deep Lake in Big Cypress National Preserve is a naturally occurring sinkhole that was formed roughly 6,000 years ago. Deep Lake has a surface area of 0.002 square miles.

1.

How many Deep Lakes are needed to cover the same surface area as Lake Okeechobee?

Learning Targets  

I can construct the circumscribed circle of a triangle.

I can locate the circumcenter of a triangle.

Lake Okeechobee averages 9 feet in depth, whereas Deep Lake averages 95 feet in depth. Based on the volume of the two lakes it would take approximately 30,850 Deep Lakes to fill up Lake Okeechobee.

2.

Explain why it takes more Deep Lakes to cover the surface area of Lake Okeechobee than to fill up Lake Okeechobee.

Unit 11: Justifying Relationships in Polygons With Circles

177

11.1.2 Exploration: Circumscribing A Triangle With A Circle

An inscribed polygon in a circle is a polygon which has all of its vertices on a circle. ∆𝑇𝑇𝑇𝑇𝑇𝑇 is inscribed in Circle 𝑅𝑅, where ̅​̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 is a radius. Lines 𝑅𝑅𝑅𝑅, 𝑅𝑅𝑅𝑅, and 𝑅𝑅𝑅𝑅 are perpendicular bisectors of ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 , ̅​̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 , and ̅​̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 , respectively.

Obtuse

Each diagram shows a different type of triangle inscribed in Circle 𝑅𝑅.

1.

Determine similarities and differences among all three inscribed triangles.

Similarities

Differences

Acute

2.

Right

178

Point 𝑅𝑅 is the center of the circle that is circumscribed about ∆𝑇𝑇𝑇𝑇𝑇𝑇. Make a conjecture about point 𝑅𝑅 as it relates to the perpendicular bisectors of the sides of ∆𝑇𝑇𝑇𝑇𝑇𝑇.

Unit 11: Justifying Relationships in Polygons With Circles

3.

Ask two other pairs of partners for the conjectures they wrote. Listen to their conjectures and write a summary.

You are the only person who should write in the first column of the table. Have the person initial next to your summary stating that your summary was accurate.

My Summary of Their Conjecture

4.

Copy ∆𝐷𝐷𝐷𝐷𝐷𝐷 onto patty paper. Construct the perpendicular bisectors of each side. Name the point of concurrency of ∆𝐷𝐷𝐷𝐷𝐷𝐷 point 𝑅𝑅.

6.

Discuss with your partner the relationship point 𝑅𝑅 has with each of the vertices.

Initials

Review your conjecture with your partner and if necessary, make any changes.

A circumscribed circle is the smallest circle that includes a plane figure. If the figure is a polygon, then the circle must contain all of the vertices of the polygon. Not every polygon has a circumscribed circle, but all triangles and all regular polygons have circumscribed circles.

5.

The point of concurrency of the perpendicular bisectors of a triangle (the center of a triangle’s circumscribed circle) is the circumcenter.

7.

Use the definition of a circumcenter to explain how the relationship point 𝑅𝑅 has with each of the vertices ensures that a circle can be drawn to circumscribe the triangle.

The point of concurrency is the point where three or more lines intersect.

Unit 11: Justifying Relationships in Polygons With Circles

179

8.

Use a compass centered at point 𝑅𝑅 to draw the circle that will pass through all three vertices.

9.

Review your conjecture. Does your conjecture hold true for newly constructed circumscribed Circle 𝑅𝑅?

11. Describe how the circumscribed circle can be used to determine the type of triangle that ∆𝐾𝐾𝐾𝐾𝐾𝐾 is. Name a property and theorem from Unit 10 that can be used to verify this.

12. The dotted line is the perpendicular bisector of side ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅.

The distance between points 𝑅𝑅 and 𝑁𝑁 is 4.35 units. If point 𝑁𝑁 is the circumcenter, what is the distance between points 𝐻𝐻 and 𝑁𝑁? Explain or show your reasoning.

10. Copy ∆𝐾𝐾𝐾𝐾𝐾𝐾 onto another piece of paper. Then, use a compass and a straight edge to construct the circle that circumscribes ∆𝐾𝐾𝐾𝐾𝐾𝐾.

180

Unit 11: Justifying Relationships in Polygons With Circles

13. Emilio tells his partner, Talia, that in order to show that point 𝑁𝑁 is the circumcenter, he only needs to construct one more perpendicular bisector. Talia challenged him to prove it. Describe how Emilio could justify that point 𝑁𝑁 is the circumcenter by drawing only one more perpendicular bisector.

11.1.3 Wrap-Up: Ticket Out the Door 1.

Describe how to construct Circle 𝐻𝐻, the circumscribed circle of ∆𝐶𝐶𝐶𝐶𝐶𝐶.

Unit 11: Justifying Relationships in Polygons With Circles

181

Unit 11, Lesson 2: Constructing the Incenter of a Triangle

11.2.1 Warm-Up: Salt Pile

Benchmark MA.912.GR.5.3 Construct the inscribed and circumscribed circles of a triangle.

Learning Targets  

I can construct an inscribed circle in a triangle.

I can locate the incenter of a triangle.

182

Unit 11: Justifying Relationships in Polygons With Circles

Salt is poured onto a triangle until it piles up. As the salt piles up, the pile reaches a maximum height. As the salt reaches a maximum height, new grains of salt fall off whichever side of the triangle is closest.

1.

11.2.2 Exploration: Constructing an Inscribed Circle 1.

∆𝑁𝑁𝑁𝑁𝑁𝑁 is shown as three different types of triangles. a. Label each point of concurrency as point 𝑅𝑅.

What do you wonder about the salt pile?

Acute

2.

What do you notice about the salt pile?

3.

What do you notice about the point where the salt reaches a maximum?

Right

Obtuse

Unit 11: Justifying Relationships in Polygons With Circles

183

b.

What do you notice about the three different triangles and point 𝑅𝑅, the point of concurrency?

The point of concurrency of the angle bisectors of a triangle is the incenter.

4.

2.

∆𝑅𝑅𝑅𝑅𝑅𝑅 is shown where each angle is bisected. The three angle bisectors meet at point 𝑇𝑇.

How is circle 𝑇𝑇 related to ∆𝑅𝑅𝑅𝑅𝑅𝑅?

3.

Consider ∆𝐾𝐾𝐾𝐾𝐾𝐾.

a.

Copy ∆𝐾𝐾𝐾𝐾𝐾𝐾 onto patty paper and find the incenter of the triangle by constructing the angle bisectors.

Label the incenter of ∆𝐾𝐾𝐾𝐾𝐾𝐾 as point 𝐻𝐻.

b.

Draw the inscribed circle for ∆𝐾𝐾𝐾𝐾𝐾𝐾 using point 𝐻𝐻, the incenter, as the center of the circle.

Use a compass to draw the inscribed circles for each type of ∆𝑁𝑁𝑁𝑁𝑁𝑁 in problem 1.

184

Unit 11: Justifying Relationships in Polygons With Circles

c.

Noah is concerned that when he draws the inscribed circle, he does not have the exact points on the triangle where the circle intersects the triangle. He drew three perpendicular lines from the center to each side.

5.

Copy ∆𝑀𝑀𝑀𝑀𝑀𝑀 onto another piece of paper. Then, use a compass and a straight edge to inscribe a circle in ∆𝑀𝑀𝑀𝑀𝑀𝑀.

6.

∆𝑁𝑁𝑁𝑁𝑁𝑁 is shown where ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃 bisects ∠𝑁𝑁𝑁𝑁𝑁𝑁 and ̅​̅​̅​̅ 𝐺𝐺𝐺𝐺 bisects ∠𝑃𝑃𝑃𝑃𝑃𝑃.

When Circle 𝐻𝐻 is drawn on Noah’s version of ∆𝐾𝐾𝐾𝐾𝐾𝐾, what relationship do ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻, ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻, and ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 have to each other and to Circle 𝐻𝐻?

d.

Draw Circle 𝐻𝐻 that passes through points 𝑇𝑇, 𝐶𝐶, and 𝑅𝑅.

a.

Will the inscribed circle of ∆𝑁𝑁𝑁𝑁𝑁𝑁 pass through points 𝑃𝑃 and 𝐺𝐺?

Unit 11: Justifying Relationships in Polygons With Circles

185

b.

Justify why ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃 and ̅​̅​̅​̅ 𝐹𝐹𝐹𝐹 are not radii of the inscribed Circle F.

111.2.3 Wrap-Up: Reflection 1.

7.

What was the muddiest (most difficult or confusing) point in today’s lesson on constructing a circle inscribed in a triangle?

Copy ∆𝐻𝐻𝐻𝐻𝐻𝐻 onto a piece of patty paper. Then, construct the incenter and circumcenter of ∆𝐻𝐻𝐻𝐻𝑊𝑊.

186

Unit 11: Justifying Relationships in Polygons With Circles

Unit 11, Lesson H1: Inscribing Polygons in Circles

11.H1.1 Warm-Up: Earth’s Orbit

Since Earth's orbit is an ellipse, the Earth is slightly closer to the sun in January. The picture shows two photos taken by the SOHO satellite.

The left side shows the sun on January 4, 2009 and the right side shows the sun on July 4, 2009.

Benchmark MA.912.GR.5.4 Construct a regular polygon inscribed in a circle. Regular polygons are limited to triangles, quadrilaterals, and hexagons.

Learning Targets   

I can construct an equilateral triangle inscribed in a circle.

I can construct a square inscribed in a circle.

I can construct a regular hexagon inscribed in a circle.

A scientist measured the two images. She found that the diameter of the image for January was 72 millimeters and the image for July was 69 millimeters.

1.

By what percentage did the diameter of the Sun change between January and July compared to its average diameter of January and July?

2.

If the average distance to the Sun from Earth is 149,600,000 kilometers, how much closer is Earth to the Sun in July compared to January?

Unit 11: Justifying Relationships in Polygons With Circles

187

4.

11.H1.2 Activity: Regular Polygons Inscribed in Circles

Make a conjecture about how to construct each regular polygon inscribed in a circle using a compass and a straight edge.

Work with your partner to answer each question. Use the larger copies of each polygon provided by your teacher.

1.

Use a compass to draw a circumscribed circle for each polygon.

2.

Mark the center of the circle that was drawn on the polygon. Draw a central angle that passes through two subsequent vertices, as shown.

Triangle

Square

Construction

Triangle

Hexagon

Square

3.

Use a protractor to determine the measure of each central angle.

Triangle

Measure of Central Angle

188

Square

Hexagon Hexagon

Unit 11: Justifying Relationships in Polygons With Circles

5.

Ask another pair of partners for the conjectures they wrote. Listen to their conjectures and write a summary.

You are the only person who should write in the first three columns of the table. Have the person initial next to your summary stating that your summary was accurate.

Other Partner Pair’s Conjecture

11.H1.3 Activity: Card Sort Sort the sets for each construction. Then, write the steps in the table.

Construction Steps

Other Partner Pair’s Initials

Triangle Method 1

Triangle

Square

Square Method 1

Hexagon

6.

Review your conjectures with your partner and if necessary, make any changes.

Unit 11: Justifying Relationships in Polygons With Circles

189

Hexagon Method 1

Triangle Method 2

190

Square Method 2

Hexagon Method 2

Unit 11: Justifying Relationships in Polygons With Circles

11.H1.4 Wrap-Up: Growth Mindset 1.

Answer each question in the table by choosing the best description of your learning today.

Which mindset did I demonstrate?

Did I use the partner and group discussions as learning opportunities?

Did I check my work and my group’s work to ensure my understanding of the constructions?

Did I put as much effort as I possibly could into the activities?

Unit 11, Lesson 3: Solving Problems Involving Triangles Inscribed in Circles

Choose the best description.

Benchmark

o Never

o Sometimes o Usually o Always

MA.912.GR.6.3 Solve mathematical problems involving triangles and quadrilaterals inscribed in a circle.

o Never

o Sometimes o Usually o Always o Never

o Sometimes o Usually o Always

Learning Target 

I can solve mathematical problems involving triangles inscribed in a circle.

Unit 11: Justifying Relationships in Polygons With Circles

191

11.3.2 Exploration: Triangles Inscribed in Circles

11.3.1 Warm-Up: Patterns

Work with your partner to complete the following.

Elijah is constructing a triangle inscribed in Circle 𝐵𝐵 and stopped after he created six points on Circle 𝐵𝐵 that are the same distance from each other.

1.

What is the first question that comes to your mind?

2.

Describe the pattern used in the image.

1.

2.

3.

Using a straightedge, draw a line from point 𝐴𝐴 to point 𝐻𝐻.

Using a straightedge, draw a line from point 𝐻𝐻 to point 𝑊𝑊.

In relation to the circle, what type of angle is ∠𝐴𝐴𝐴𝐴𝐴𝐴?

192

Unit 11: Justifying Relationships in Polygons With Circles

4.

⏜.

Explain the relationship between 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 and 𝑚𝑚 𝑊𝑊𝑊𝑊

11.3.3 Guided Practice: Triangles Inscribed in Circles 1.

5.

6.

7.

8.

Using a straightedge, draw a line from point 𝐴𝐴 to point 𝑊𝑊.

Circle 𝑅𝑅 is shown. Points 𝑃𝑃, 𝐶𝐶, and 𝐺𝐺 are on circle 𝑅𝑅 and ⏜ = 162°, 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 58°, and create ∆𝑃𝑃𝑃𝑃𝑃𝑃. 𝑚𝑚 𝑃𝑃𝑃𝑃 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 = (7𝑥𝑥 − 17)°.

Use a ruler or patty paper to verify that ∆𝐴𝐴𝐴𝐴𝐴𝐴 is an equilateral triangle.

What is the 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴?

⏜?

What is the 𝑚𝑚 𝑊𝑊𝑊𝑊

a.

What is the 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃?

b.

Find the value of 𝑥𝑥.

c.

What is the 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃?

Unit 11: Justifying Relationships in Polygons With Circles

193

2.

Circle 𝑁𝑁 has points 𝐻𝐻, 𝑊𝑊, and 𝐵𝐵 on the circle and ⏜ = 102° diameter ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 . ∆𝐻𝐻𝐻𝐻𝐻𝐻 is inscribed in circle 𝑁𝑁. 𝑚𝑚 𝐵𝐵𝐵𝐵 and 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 = (4𝑥𝑥 − 10)°.

a.

3.

∆𝐶𝐶𝐶𝐶𝐶𝐶 and ∆𝐹𝐹𝐹𝐹𝐹𝐹 are inscribed in Circle P, where ̅​̅​̅​̅​̅ 𝐹𝐹𝐹𝐹 is ⏜ the diameter of Circle 𝑃𝑃. 𝑚𝑚 𝐹𝐹𝐹𝐹 = (2𝑥𝑥 − 6)°, ⏜ = (4𝑥𝑥 − 6)°, and 𝑚𝑚 𝐿𝐿𝐿𝐿 ⏜ = (3𝑥𝑥 + 30)°. 𝑚𝑚 𝐿𝐿𝐿𝐿

What is the 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻?

Complete the table.

b.

194

Find the value of 𝑥𝑥.

Angle Measures

Angle Measures

𝑚𝑚∠𝐿𝐿𝐿𝐿𝐿𝐿 =

𝑚𝑚∠𝐿𝐿𝐿𝐿𝐿𝐿 =

𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹 =

Unit 11: Justifying Relationships in Polygons With Circles

𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹 = 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹 =

2.

11.3.4 Your Turn!

1.

Circle 𝐾𝐾 has points 𝑉𝑉, 𝐶𝐶, and 𝑍𝑍 on the circle, and ∆𝑉𝑉𝑉𝑉𝑉𝑉 is ⏜ = (9𝑥𝑥 + 23)°, 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 36°, and inscribed in circle 𝐾𝐾. 𝑚𝑚 𝑉𝑉𝑉𝑉 𝑚𝑚∠𝑉𝑉𝑉𝑉𝑉𝑉 = (7𝑥𝑥 − 21)°.

a.

What is the value of 𝑥𝑥?

Circle 𝐴𝐴 has points 𝑁𝑁, 𝐷𝐷, 𝐵𝐵, and 𝑆𝑆 on the circle, and ⏜ = (7𝑥𝑥 − 7)°, ∆𝑆𝑆𝑆𝑆𝑆𝑆 is inscribed in Circle 𝐴𝐴. 𝑚𝑚 𝑁𝑁𝑁𝑁 ⏜ = (23𝑦𝑦 − 5)°, 𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁 = (5𝑥𝑥 − 23)°, and 𝑚𝑚 𝐷𝐷𝐷𝐷𝐷𝐷 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = (8𝑦𝑦 + 29)°.

Complete the table.

Angle Measures 𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁 = 𝑚𝑚∠𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁 =

b.

Arc Measures ⏜ = 𝑚𝑚 𝑁𝑁𝑁𝑁 ⏜= 𝑚𝑚 𝐷𝐷𝐷𝐷𝐷𝐷 ⏜ = 𝑚𝑚 𝑁𝑁𝑁𝑁

What is the 𝑚𝑚∠𝑉𝑉𝑉𝑉𝑉𝑉?

Unit 11: Justifying Relationships in Polygons With Circles

195

3.

Circle 𝐺𝐺 has diameter ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 and ∆𝑀𝑀𝑀𝑀𝑀𝑀 inscribed in the circle. 𝑚𝑚∠𝑇𝑇𝑇𝑇𝑇𝑇 = (6𝑥𝑥 − 8)° and 𝑚𝑚∠𝑇𝑇𝑇𝑇𝑇𝑇 = (4𝑥𝑥 + 13)°.

a.

Find the value of 𝑥𝑥.

b.

What is the 𝑚𝑚∠𝑇𝑇𝑇𝑇𝑇𝑇?

c.

What is the 𝑚𝑚∠𝑇𝑇𝑇𝑇𝑇𝑇?

196

11.3.5 Wrap-Up: Reflection 1.

Today I am still unsure about …

2.

To fill in this gap, I intend to …

Unit 11: Justifying Relationships in Polygons With Circles

Unit 11, Lesson 4: Solving Problems Involving Quadrilaterals Inscribed in Circles Benchmark

MA.912.GR.6.3 Solve mathematical problems involving triangles and quadrilaterals inscribed in a circle.

11.4.1 Warm-Up: Notice and Wonder Machu Picchu is an Inca citadel in southern Peru. Machu Picchu was built in the classical Inca style with polished drystone walls and consists of more than 150 buildings.

Although many of the stones that were used to build the city were more than 50 pounds, it is believed that no wheels were used to transport these rocks up the mountain.

Learning Target 

I can solve mathematical problems involving quadrilaterals inscribed in a circle.

1.

What do you notice?

2.

What do you wonder?

Unit 11: Justifying Relationships in Polygons With Circles

197

c.

11.4.2 Exploration: Quadrilaterals Inscribed in Circles 1.

⏜ = 100°, Circle 𝐶𝐶 is shown, where ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 ≅ ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷. 𝑚𝑚 𝑅𝑅𝑅𝑅 ⏜ = 136°, and 𝑚𝑚 𝐻𝐻𝐻𝐻𝐻𝐻 ⏜ = 224°. ⏜ = 68°, 𝑚𝑚 𝐻𝐻𝐻𝐻𝐻𝐻 𝑚𝑚 𝐷𝐷𝐷𝐷

Ask two classmates for their conjectures. Record their conjectures and write your summary of their reason. You are the only person who should write in the first two columns of the table. Have the person initial next to your summary stating that your summary was correct.

Conjecture

a. Complete the table.

Angle Measures 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 =

𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = b.

198

𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 =

d.

My Summary of Their Reason

Initials

Review your conjecture with your partner. If necessary, make any changes.

Angle Measures 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 = 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 =

Work with your partner to make a conjecture about the relationships of the angles in 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷.

Unit 11: Justifying Relationships in Polygons With Circles

2.

Circle 𝑁𝑁 has points 𝐻𝐻, 𝐴𝐴, 𝑆𝑆, and 𝑊𝑊 on the circle.

Quadrilateral 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 is inscribed in the circle. ⏜ = 59°, 𝑚𝑚 𝐻𝐻𝐻𝐻𝐻𝐻 ⏜ ⏜ 𝑚𝑚 𝐻𝐻𝐻𝐻 = 166° and 𝑚𝑚 𝑊𝑊𝑊𝑊𝑊𝑊 = 136°.

11.4.3 Guided Practice: Quadrilaterals Inscribed in Circles 1.

Complete the statement.

If a quadrilateral is inscribed in a circle, angles are

2.

Use your conjecture to find the measures.

Angle Measures 𝑚𝑚∠𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 =

𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 =

𝑚𝑚∠𝑆𝑆𝑆𝑆𝑆𝑆 =

Arc Measures ⏜= 𝑚𝑚 𝑆𝑆𝑆𝑆

o adjacent o opposite

o congruent. o complementary. o supplementary.

Circle 𝑃𝑃 has 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 inscribed in the circle. ⏜ = 206° and 𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇 𝑚𝑚∠𝐺𝐺𝐺𝐺𝐺𝐺 = 84°. a.

What is the 𝑚𝑚∠𝑇𝑇𝑇𝑇𝑇𝑇?

b.

What is the 𝑚𝑚∠𝐺𝐺𝐺𝐺𝐺𝐺?

⏜= 𝑚𝑚 𝑆𝑆𝑆𝑆 ⏜= 𝑚𝑚 𝐻𝐻𝐻𝐻

Unit 11: Justifying Relationships in Polygons With Circles

199

3.

Circle 𝑅𝑅 has 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 inscribed in the circle. 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 = (2𝑥𝑥 − 7)°, 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 = (2𝑥𝑥 + 3)°, and 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 = (𝑥𝑥 + 4)°. a.

b.

Find the value of 𝑥𝑥.

11.4.4 Your Turn!

1.

Circle 𝐴𝐴 has 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 inscribed in the circle. 𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁 = (6𝑥𝑥 + 32)°, 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹 = (6𝑦𝑦 − 8)°, 𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁 = 73°, and 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹 = (3𝑦𝑦 − 1)°. a.

Find the value of 𝑥𝑥.

b.

What is the 𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁?

c.

Find the value of 𝑦𝑦.

d.

What is the 𝑚𝑚∠𝐹𝐹𝐹𝐹𝐹𝐹?

Complete the table.

𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 = 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 = 𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 =

𝑚𝑚∠𝐸𝐸𝐸𝐸𝐸𝐸 =

200

Unit 11: Justifying Relationships in Polygons With Circles

2.

⏜ = 71°, Circle 𝐸𝐸 has 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 inscribed in the circle. 𝑚𝑚 𝑇𝑇𝑇𝑇 ⏜ = 61°, and 𝑚𝑚∠𝑇𝑇𝑇𝑇𝑇𝑇 = (12𝑥𝑥 − 3)°. 𝑚𝑚 𝑀𝑀𝑀𝑀

11.4.5 Wrap-Up: Error Analysis 1.

𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 is inscribed in Circle 𝐾𝐾, where 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 = (3𝑥𝑥 + 23)° and 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐴𝐴 = (6𝑥𝑥 + 13)°.

Anastasia is finding the value of 𝑥𝑥. Her work is shown.

Anastasia’s Work

a.

b.

Find the value of 𝑥𝑥.

3𝑥𝑥 + 23 = 6𝑥𝑥 + 13 3𝑥𝑥 = 10 10 𝑥𝑥 = 3

a.

Write a note to Anastasia explaining her mistake.

b.

Find the value of 𝑥𝑥.

What is the 𝑚𝑚∠𝑇𝑇𝑇𝑇𝑇𝑇?

Unit 11: Justifying Relationships in Polygons With Circles

201

Unit 11, Lesson 5: Interior and Exterior Angles of Triangles

11.5.1 Warm-Up: Superhero 1.

If geometry had a superpower, it would be …

Benchmark

MA.912.GR.1.3 Prove relationships and theorems about triangles. Solve mathematical and real-world problems involving postulates, relationships, and theorems of triangles.

because …

Learning Targets   

I can show that the sum of the interior angles of a triangle is 180 degrees.

I can show that the sum of the exterior angles of a triangle is 360 degrees.

I can solve problems involving postulates, relationships, and theorems of triangles.

2.

Geometry’s villain would be …

because …

202

Unit 11: Justifying Relationships in Polygons With Circles

11.5.2 Exploration: Interior Angles of Triangles 1.

Circle 𝐴𝐴 is shown, where ⏜ = 102°, 𝑚𝑚 𝐵𝐵𝐵𝐵 ⏜ = 108°, 𝑚𝑚 𝑅𝑅𝑅𝑅 ⏜ = 150°. and 𝑚𝑚 𝐻𝐻𝐻𝐻

a.

Complete the table.

Angle Measures

2.

Circle 𝐸𝐸 is shown where points 𝑃𝑃, 𝐶𝐶, and 𝑀𝑀 are on the circle. ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 and ̅​̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 are tangent to Circle 𝐸𝐸 and intersect at point 𝑅𝑅 ̅​̅​̅​̅ and outside the circle. 𝐺𝐺𝐺𝐺 ̅​̅​̅​̅​̅ 𝐺𝐺𝐺𝐺 are tangent to Circle 𝐸𝐸 and intersect at point 𝐺𝐺 outside the circle. ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 and ̅​̅​̅​̅​̅ are tangent to Circle 𝐸𝐸 𝑊𝑊𝑊𝑊 and intersect at point 𝑊𝑊 outside the circle. ⏜ = 108°, 𝑚𝑚 ⏜ 𝐶𝐶𝐶𝐶 = 136°, 𝑚𝑚 𝑀𝑀𝑀𝑀 ⏜ = 116°. and 𝑚𝑚 𝐶𝐶𝐶𝐶 a.

𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 =

Complete the table.

Angle Measures

𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 =

𝑚𝑚∠𝑊𝑊𝑊𝑊𝑊𝑊 =

𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 =

𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 = 𝑚𝑚∠𝐺𝐺𝐺𝐺𝐺𝐺 =

b.

What is the sum of the angles in ∆𝑅𝑅𝑅𝑅𝑅𝑅?

b.

What is the sum of the angles in ∆𝑅𝑅𝑅𝑅𝑅𝑅?

c.

Write a conjecture about the sum of the interior angles of a triangle.

Unit 11: Justifying Relationships in Polygons With Circles

203

d.

Ask two classmates for their conjectures. Record their conjectures and write your summary of their reason. You are the only person who should write in the first two columns of the table. Have the person initial next to your summary stating that your summary was correct.

Conjecture

e.

My Summary of Their Reason

11.5.3 Activity: Exterior Angles of Triangles Work with your partner to complete the following.

1.

On a separate sheet of paper, draw three line segments, ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵, ̅​̅​̅​̅ 𝑉𝑉𝑉𝑉, and ̅​̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 , to make ∆𝑉𝑉𝑉𝑉𝑉𝑉 as shown.

Initials

Review your conjecture with your partner. If necessary, make any changes.

204

Triangle Sum Theorem

An exterior angle is an angle created by extending one leg of a triangle. a.

List the exterior angles of ∆𝑉𝑉𝑉𝑉𝑉𝑉.

b.

Estimate the sum of the exterior angles of ∆𝑉𝑉𝑉𝑉𝑉𝑉.

The sum of the measures of the interior angles of a triangle is 180°.

Unit 11: Justifying Relationships in Polygons With Circles

2.

d.

⏜ using point 𝑊𝑊 as the Use a compass to draw arc 𝐵𝐵𝐵𝐵 ⏜ using ⏜ using point 𝐵𝐵 as the center, and 𝑁𝑁𝑁𝑁 center, 𝐾𝐾𝐾𝐾 point 𝑉𝑉 as the center, as shown.

Ask two classmates for their conjectures. Record their conjectures and write your summary of their reason. You are the only person who should write in the first two columns of the table. Have the person initial next to your summary stating that your summary was correct.

Conjecture

a.

Trace the figure above onto patty paper and cut out each sector. Rearrange the sectors to determine the sum of the exterior angles.

b.

What do you notice about the sum of the exterior angles of the triangle?

c.

Write a conjecture about the sum of the exterior angles of a triangle.

e.

My Summary of Their Reason

Initials

Review your conjecture with your partner. If necessary, make any changes.

Unit 11: Justifying Relationships in Polygons With Circles

205

11.5.4 Guided Practice: Interior and Exterior Angles of Triangles 1.

∆𝑅𝑅𝑅𝑅𝑅𝑅 is shown where 𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁 = 14𝑥𝑥 + 3, 9 5 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 = 2 𝑥𝑥 − 2, and 𝑚𝑚∠𝑊𝑊𝑊𝑊𝑊𝑊 = 9𝑥𝑥 − 13.

a.

b.

Find the value of 𝑥𝑥.

Complete the table.

𝑚𝑚∠𝑁𝑁𝑁𝑁𝑁𝑁 =

206

𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 =

2.

∆𝑃𝑃𝑃𝑃𝑃𝑃 is shown, where ∠𝑆𝑆𝑆𝑆𝑆𝑆, ∠𝑃𝑃𝑃𝑃𝑃𝑃, and ∠𝐻𝐻𝐻𝐻𝐻𝐻 are exterior angles. 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑃𝑃 = 52°, 𝑚𝑚∠𝑆𝑆𝑆𝑆𝑆𝑆 = (11𝑥𝑥 − 20)°, and 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 = (8𝑥𝑥 + 5)°.

a.

Find the value of 𝑥𝑥.

b.

What is the measure of ∠𝑆𝑆𝑆𝑆𝑆𝑆?

c.

What is the measure of ∠𝐻𝐻𝐻𝐻𝐻𝐻?

𝑚𝑚∠𝑊𝑊𝑊𝑊𝑊𝑊 =

Unit 11: Justifying Relationships in Polygons With Circles

11.5.5 Wrap-Up: Reflection 1.

Choose the emoji that best reflects your understanding of the sum of interior and exterior angles of triangles.

Unit 11, Lesson 6: Triangle Inequality Theorem Benchmarks

MA.912.GR.1.3 Prove relationships and theorems about triangles. Solve mathematical and real-world problems involving postulates, relationships, and theorems of triangles.

MA.912.LT.4.10 Judge the validity of arguments and give counterexamples to disprove statements.

Learning Targets  

I can determine if a set of triangle side lengths can be used to construct a triangle.

I can use the triangle inequality theorem to prove relationships in triangles.

Unit 11: Justifying Relationships in Polygons With Circles

207

11.6.2 Exploration: Triangle Inequality Theorem

11.6.1 Warm-Up: Number Riddle 1.

Use any of the four operations (+, -, ×, ÷) with the numbers shown to produce a result of 𝟑𝟑𝟑𝟑.

7, 0, 2, 3, 5

208

1.

With your partner, use a ruler to determine if the given measurements can be used to create a triangle. Then, complete the table.

Measurements in Centimeters (cm)

Can it create a triangle?

3 cm, 6 cm, and 8 cm

o Yes o No

8 cm, 11 cm, and 15 cm

o Yes o No

4 cm, 5 cm, and 11 cm

o Yes o No

3 cm, 4 cm, and 10 cm

o Yes o No

2 cm, 5 cm, and 8 cm

o Yes o No

Explanation

2.

What similarities did you notice about the measurements that could create a triangle?

3.

What similarities did you notice about the measurements that could not create a triangle?

Unit 11: Justifying Relationships in Polygons With Circles

4.

5.

Make a conjecture about how to determine if side lengths can be used to construct a triangle if you cannot draw it.

Ask two other pairs of partners for the conjectures they wrote. Listen to their conjectures and write a summary.

You are the only person who should write in the first column of the table. Have the person initial next to your summary stating that your summary was accurate.

My Summary of the Conjecture

6.

7.

8.

Check your answer by constructing the triangle with a ruler.

9. What did you notice about the measurements 3 cm, 4 cm, and 7 cm?

10. Adjust your conjecture if needed.

Initials

Review your conjecture with your partner and if necessary, make any changes.

Use your conjecture to determine if 3 cm, 4 cm, and 7 cm will create a triangle.

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

If three segments have lengths such that the sum of any two is greater than the third, then a triangle can be constructed from the segments.

Unit 11: Justifying Relationships in Polygons With Circles

209

11.6.4 Exploration: Relating Angle Measures and Sides of Triangles

11.6.3 Try It: Triangle Inequality Theorem 1.

Complete the table.

Side Lengths 57, 16, and 75 33, 12, and 25 21, 5, and 26 61, 100, and 61 48, 24, and 21

210

Work

Can it create a triangle?

∆𝑅𝑅𝑅𝑅𝑅𝑅 is shown, where 𝑊𝑊𝑊𝑊 = 24, 𝐶𝐶𝐶𝐶 = 18, and 𝑅𝑅𝑅𝑅 = 10.

o Yes o No o Yes o No o Yes o No

1.

o Yes o No

Use a protractor to find the angle measures. Record your answers in the table.

Angle Measures 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 =

o Yes o No

𝑚𝑚∠𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑚𝑚∠𝑅𝑅𝑅𝑅𝑅𝑅 =

2.

List the side lengths in order from longest to shortest.

3.

List the angles in order from largest to smallest.

Unit 11: Justifying Relationships in Polygons With Circles

4.

What do you notice about the size of the angle compared to the length of the opposite side?

11.6.5 Guided Instruction: Proving Inequality in a Triangle

If two sides of a triangle are unequal in measure, then the angle opposite the longer side has the greater measure.

If two angles of a triangle are unequal in measure, then the side opposite the angle with the greater measure is the longer side.

Whole Greater Than Parts Theorem

If 𝑎𝑎 > 0, 𝑏𝑏 > 0, and 𝑎𝑎 + 𝑏𝑏 = 𝑐𝑐, then 𝑐𝑐 > 𝑎𝑎 and 𝑐𝑐 > 𝑏𝑏.

Unit 11: Justifying Relationships in Polygons With Circles

211

1.

∆𝑇𝑇𝑇𝑇𝑇𝑇 is shown, with ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 creating ∆𝐻𝐻𝐻𝐻𝐻𝐻 and ∆𝐷𝐷𝐷𝐷𝐷𝐷, where ∆𝐷𝐷𝐷𝐷𝐷𝐷 is an equilateral triangle.

11.6.6 Try It: Triangle’s Angle Measures 1.

∆𝑇𝑇𝑇𝑇𝑇𝑇 is shown, where 𝐾𝐾𝐾𝐾 = 32, 𝑀𝑀𝑀𝑀 = 34, and 𝑇𝑇𝑇𝑇 = 13.

List the angles in order from least to greatest.

Given: ∆𝐷𝐷𝐷𝐷𝐷𝐷 is equilateral; 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = 22° Prove: 𝐻𝐻𝐻𝐻 > 𝐻𝐻𝐻𝐻

Statement 1. ∆𝐷𝐷𝐷𝐷𝐷𝐷 is equilateral; 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = 22° 2.

Reason 1. Given

3. 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 + 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = 𝑚𝑚∠𝑀𝑀𝑀𝑀𝑀𝑀

3.

4.

4. Substitution Property of Equality

5. 𝑚𝑚∠𝑀𝑀𝑀𝑀𝑀𝑀 > 𝑚𝑚∠𝑇𝑇𝑇𝑇𝑇𝑇

5. Whole greater than parts theorem

6. 𝐻𝐻𝐻𝐻 > 𝐻𝐻𝐻𝐻

6.

212

Unit 11: Justifying Relationships in Polygons With Circles

2. Property of equilateral triangles

11.6.7 Wrap-Up: Reflection 1.

2.

One thing I struggled with today was …

One thing I need to practice more is …

Unit 11, Lesson 7: Centroid of a Triangle Benchmarks

MA.912.GR.1.3 Prove relationships and theorems about triangles. Solve mathematical and real-world problems involving postulates, relationships, and theorems of triangles.

MA.912.GR.5.2 Construct the bisector of a segment or an angle, including the perpendicular bisector of a line segment.

Learning Targets

3.

One thing I understand very well is …

 

I can find the centroid of a triangle.

I can use the relationship of the distance a centroid is from a vertex to the midpoint of the opposite side to solve problems.

Unit 11: Justifying Relationships in Polygons With Circles

213

11.7.1 Warm-Up: Bell Work 1.

Given ∆𝐵𝐵𝐵𝐵𝐵𝐵, where 𝑀𝑀𝑀𝑀 = 7, 𝑀𝑀𝑀𝑀 = 12, and 𝐵𝐵𝐵𝐵 = 9, list the angles in order from least to greatest.

11.7.2 Exploration: Centroid of a Triangle 1.

All polygons have a balancing point. The diagrams show the balancing point of a triangle.

a.

214

What do you notice?

Unit 11: Justifying Relationships in Polygons With Circles

b.

Discuss with your partner how you think the balance point was found. Summarize your discussion.

3.

∆𝐻𝐻𝐻𝐻𝐻𝐻 is shown.

a.

2.

A median of a triangle is a segment that joins any vertex of the triangle to the midpoint of the opposite side. ∆𝐾𝐾𝐾𝐾𝐾𝐾 is shown, where point 𝐺𝐺 is a point of concurrency.

Discuss with your partner how to find the midpoints of the triangle’s sides using each method. Complete the table.

Patty Paper Compass

Use the steps to construct the centroid of ∆𝐻𝐻𝐻𝐻𝐻𝐻.

Step 1: Trace ∆𝐻𝐻𝐻𝐻𝐻𝐻 on patty paper.

̅​̅​̅​̅ to find Step 2: Construct the perpendicular bisector of 𝐻𝐻𝐻𝐻 its midpoint, 𝑀𝑀.

Step 3: Draw the median from 𝑀𝑀 to the opposite vertex, ∠𝐻𝐻𝐻𝐻𝐻𝐻. ̅​̅​̅​̅ to find Step 4: Construct the perpendicular bisector of 𝐻𝐻𝐻𝐻 its midpoint, 𝑅𝑅.

Step 5: Draw the median from 𝑅𝑅 to the opposite vertex, ∠𝐻𝐻𝐻𝐻𝐻𝐻. ̅​̅​̅​̅ to find Step 6: Construct the perpendicular bisector of 𝑇𝑇𝑇𝑇 its midpoint, 𝑋𝑋.

Step 7: Draw the median from midpoint 𝑋𝑋 to the opposite vertex, ∠𝑇𝑇𝑇𝑇𝑇𝑇.

Step 8: Draw point 𝑊𝑊, the point of concurrency, where medians ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅, ̅​̅​̅​̅ 𝑋𝑋𝑋𝑋, and ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 intersect. b.

What do you notice about the location of point 𝐶𝐶 in ∆𝐻𝐻𝐻𝐻𝐻𝐻?

Unit 11: Justifying Relationships in Polygons With Circles

215

4.

∆𝑌𝑌𝑌𝑌𝑌𝑌 is shown, with medians ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 , ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 , and ̅​̅​̅​̅ 𝑌𝑌𝑌𝑌 . Point 𝑃𝑃 ̅​̅​̅​̅, is the midpoint of ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 , point 𝑅𝑅 is the midpoint of 𝑇𝑇𝑇𝑇 ̅​̅​̅​̅ and Point 𝐾𝐾 is the midpoint of 𝑌𝑌𝑌𝑌 .

The centroid of a triangle is the intersection of the three medians of the triangle (each median connecting a vertex with the midpoint of the opposite side). b.

a.

Write a conjecture about the distance a centroid is from the vertex and the distance a centroid is from the midpoint of the opposite side.

Copy the first segment using patty paper. Use the copy to determine the relationship the first segment and the point have with the second segment.

Complete the table.

First Segment

Point

Second Segment

̅​̅​̅​̅ 𝐷𝐷𝐷𝐷

𝑅𝑅

̅​̅​̅​̅ 𝑇𝑇𝑇𝑇

̅​̅​̅​̅ 𝐹𝐹𝐹𝐹

𝐾𝐾

̅​̅​̅​̅ 𝑌𝑌𝑌𝑌

̅​̅​̅​̅ 𝐺𝐺𝐺𝐺

𝑃𝑃

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊

216

Relationship

Unit 11: Justifying Relationships in Polygons With Circles

11.7.3 Guided Instruction: Centroid of a Triangle The length from the vertex to the centroid is twice the length from the centroid to the midpoint of the opposite side, which yields a ratio of 2: 1.

1.

2.

Triangle 𝑀𝑀𝑀𝑀𝑀𝑀 is shown, where point 𝑅𝑅 is the centroid of the triangle. 𝑀𝑀𝑀𝑀 = (5𝑥𝑥 − 9), 𝑅𝑅𝑅𝑅 = (3𝑤𝑤 + 7), 𝑅𝑅𝑅𝑅 = (12𝑦𝑦 − 9), 𝑅𝑅𝑅𝑅 = 1.75𝑥𝑥, 𝑅𝑅𝑅𝑅 = (4𝑦𝑦 + 2), and 𝐻𝐻𝐻𝐻 = (𝑤𝑤 + 9).

∆𝑅𝑅𝑅𝑅𝑅𝑅 is shown, where point 𝐻𝐻 is the centroid of the triangle. 𝐷𝐷𝐷𝐷 = 11.3, 𝑅𝑅𝑅𝑅 = 20.55, and 𝐵𝐵𝐵𝐵 = 15.2.

a.

Find the value of each variable.

𝑥𝑥 = a.

𝑤𝑤 =

Complete the table.

𝐻𝐻𝐻𝐻 =

𝐷𝐷𝐷𝐷 =

𝐻𝐻𝐻𝐻 =

𝐵𝐵𝐵𝐵 =

𝑅𝑅𝑅𝑅 = b.

𝑦𝑦 =

𝐻𝐻𝐻𝐻 =

b.

If 𝐵𝐵𝐵𝐵 = (6𝑥𝑥 + 3) and 𝐾𝐾𝐾𝐾 = (10𝑥𝑥 − 7), what is the length of ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵?

Complete the table.

𝐻𝐻𝐻𝐻 =

𝑅𝑅𝑅𝑅 =

𝐻𝐻𝐻𝐻 =

𝑅𝑅𝑅𝑅 =

𝑀𝑀𝑀𝑀 =

𝑀𝑀𝑀𝑀 =

𝑅𝑅𝑅𝑅 =

𝑉𝑉𝑉𝑉 =

Unit 11: Justifying Relationships in Polygons With Circles

𝑉𝑉𝑉𝑉 =

217

c.

11.7.4 Your Turn!

1.

∆𝐶𝐶𝐶𝐶𝐶𝐶 is shown, where point 𝐵𝐵 is the centroid of the 1 circle. 𝑇𝑇𝑇𝑇 = ( 𝑥𝑥 − 4), 𝑃𝑃𝑃𝑃 = 4.4, 𝐶𝐶𝐶𝐶 = (𝑦𝑦 + 6), 2

𝐵𝐵𝐵𝐵 = (2𝑥𝑥 − 31), and 𝐵𝐵𝐵𝐵 = ( 𝑦𝑦 + 5).

3

d.

What is the value of 𝑥𝑥?

b.

What is the value of 𝑦𝑦?

218

𝑇𝑇𝑇𝑇 =

𝐵𝐵𝐵𝐵 =

𝐵𝐵𝐵𝐵 =

𝑃𝑃𝑃𝑃 =

𝐶𝐶𝐶𝐶 =

1

a.

Complete the table.

𝐶𝐶𝐶𝐶 =

If 𝐶𝐶𝐶𝐶 = (4𝑥𝑥 − 10) and 𝑇𝑇𝑇𝑇 = (2𝑥𝑥 + 1), what is the length of 𝐶𝐶𝐶𝐶?

Unit 11: Justifying Relationships in Polygons With Circles

2.

∆𝐷𝐷𝐷𝐷𝐷𝐷 is shown, where point 𝑆𝑆 is the centroid.

11.7.5 Wrap-Up: Cool Down 1.

a.

Write a short note summarizing today’s lesson to a friend who missed class.

Leonardo is finding the value of 𝑥𝑥 using the 1 information 𝑉𝑉𝑉𝑉 = ( 𝑥𝑥 + 1) and 𝑆𝑆𝑆𝑆 = (2𝑥𝑥 − 6). His work 2 is shown.

Leonardo’s Work 1 2(2𝑥𝑥 − 6) = 𝑥𝑥 + 1 2 1 4𝑥𝑥 − 12 = 𝑥𝑥 + 1 2 7 𝑥𝑥 = 13 2 26 𝑥𝑥 = 7

b.

Write a note to Leonardo explaining his mistake.

c.

What is the value of 𝑥𝑥?

Unit 11: Justifying Relationships in Polygons With Circles

219

Unit 11, Lesson 8: Triangle Midsegment Theorem

11.8.1 Warm-Up: “Many Small Cubes” 1.

Benchmarks

Japanese architect Sou Fujimoto exhibited a sculpture of stacked and suspended aluminum cubes as a part of the art fair in the Parisian Jardins des Tuileries’ gardens.

MA.912.GR.1.3 Prove relationships and theorems about triangles. Solve mathematical and real-world problems involving postulates, relationships, and theorems of triangles.

MA.912.LT.4.10 Judge the validity of arguments and give counterexamples to disprove statements.

Learning Targets  

I can discover the triangle midsegment theorem using AA similarity.

I can use the triangle midsegment theorem to prove relationships and to solve problems.

a.

What do you notice about the structure?

b.

What do you wonder?

220

Unit 11: Justifying Relationships in Polygons With Circles

11.8.2 Exploration: Triangle Midsegment Theorem ∆𝑁𝑁𝑁𝑁𝑁𝑁 and ∆𝑁𝑁𝑁𝑁𝑁𝑁 are shown, where ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 ∥ ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅.

3.

Given: ∆𝑁𝑁𝑁𝑁𝑁𝑁 with ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 ∥ ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 Prove: ∆𝑁𝑁𝑁𝑁𝑁𝑁~∆𝑁𝑁𝑁𝑁𝑁𝑁

Complete the flowchart proof.

1.

Discuss with your partner what can be concluded from the given information. Justify your conclusions.

2.

Ask a classmate for two conclusions they made. Record their conclusions and write your summary of their reason. You are the only person who should write in the first two columns of the table. Have the person initial next to your summary stating that your summary was correct.

Conclusion

My Summary of Their Reason

Initials

Unit 11: Justifying Relationships in Polygons With Circles

221

4.

What other information about triangles 𝑁𝑁𝑁𝑁𝑁𝑁 and 𝑁𝑁𝑁𝑁𝑁𝑁 can be determined from knowing that ∆𝑁𝑁𝑁𝑁𝑁𝑁~∆𝑁𝑁𝑁𝑁𝑁𝑁?

5.

Complete the table using the information 𝐾𝐾𝐾𝐾 = 5, 𝑁𝑁𝑁𝑁 = 6, 𝑁𝑁𝑁𝑁 = 3, and 𝑁𝑁𝑁𝑁 = 8.8.

𝑅𝑅𝑅𝑅 =

𝐾𝐾𝐾𝐾 =

𝑁𝑁𝑁𝑁 = 𝑉𝑉𝑉𝑉 =

̅​̅​̅​̅?

What do you notice about the location of 𝐾𝐾𝐾𝐾

6.

̅​̅​̅​̅ and 𝑅𝑅𝑅𝑅 ̅​̅​̅​̅?

What do you notice about the lengths of 𝐾𝐾𝐾𝐾

7.

222

11.8.3 Guided Practice: Triangle Midsegment Theorem 1.

∆𝑀𝑀𝑀𝑀𝑀𝑀 is shown, where ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 is the midsegment.

1

𝑀𝑀𝑀𝑀 = (2 𝑥𝑥 + 4),

𝐺𝐺𝐺𝐺 = (11𝑧𝑧 − 2), 𝑆𝑆𝑆𝑆 = (2𝑦𝑦 + 5), 𝑀𝑀𝑀𝑀 = (8𝑦𝑦 − 2), 3 1 𝑆𝑆𝑆𝑆 = (4 𝑥𝑥 − 4), 𝑊𝑊𝑊𝑊 = (9𝑧𝑧 + 1). a.

Find the value of 𝑥𝑥.

b.

Find the value of 𝑦𝑦.

c.

Find the value of 𝑧𝑧.

d.

What is the perimeter of ∆𝑀𝑀𝑀𝑀𝑊𝑊?

Triangle Midsegment Theorem The midsegment of a triangle joins the midpoints of two sides of a triangle such that its length is half the length of the third side of the triangle, and it is parallel to the third side of the triangle.

Unit 11: Justifying Relationships in Polygons With Circles

2.

∆𝐵𝐵𝐵𝐵𝐵𝐵 is shown, where 𝐷𝐷, 𝑃𝑃, and 𝐻𝐻 are the midpoints and the perimeter of ∆𝐷𝐷𝐷𝐷𝐷𝐷 is 23.5. 𝐷𝐷𝐷𝐷 = 17𝑥𝑥, 𝑃𝑃𝑃𝑃 = (6𝑥𝑥 + 5), and 𝐻𝐻𝐻𝐻 = 20𝑥𝑥 − 3.

a.

3.

∆𝑆𝑆𝑆𝑆𝑆𝑆 and ∆𝐻𝐻𝐻𝐻𝐻𝐻 are shown.

Given: 𝑆𝑆𝑆𝑆 = 𝐻𝐻𝐻𝐻 and 𝐺𝐺𝐺𝐺 = 𝑇𝑇𝑇𝑇 Prove: ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 ∥ ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

What is the value of 𝑥𝑥?

Since 𝑆𝑆𝑆𝑆 = 𝐻𝐻𝐻𝐻 and 𝐺𝐺𝐺𝐺 = 𝑇𝑇𝑇𝑇, point 𝐻𝐻 is the ___________ of ̅​̅​̅ 𝑆𝑆𝑆𝑆̅ and point 𝑇𝑇 is the ____________ of ̅​̅​̅​̅ 𝐺𝐺𝐺𝐺

b.

̅​̅​̅​̅ is the by_______________________. Therefore, 𝐻𝐻𝐻𝐻

̅​̅​̅​̅ by the _____________ of ∆𝑆𝑆𝑆𝑆𝑆𝑆 by definition and ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 ∥ 𝐻𝐻𝐻𝐻

Complete the table. 𝐵𝐵𝐵𝐵 = 𝐵𝐵𝐵𝐵 =

𝐷𝐷𝐷𝐷 =

____________________. 𝐷𝐷𝐷𝐷 = 𝐾𝐾𝐾𝐾 =

𝐾𝐾𝐾𝐾 =

Unit 11: Justifying Relationships in Polygons With Circles

223

11.8.4 Your Turn!

11.8.5 Wrap-Up: Error Analysis

∆𝐾𝐾𝐾𝐾𝐾𝐾 is shown, where 𝑇𝑇, 𝑉𝑉, and 𝐻𝐻 are midpoints.

1 𝐾𝐾𝐾𝐾 = (5𝑥𝑥), 𝑉𝑉𝑉𝑉 = ( 𝑦𝑦 + 11), 𝑉𝑉𝑉𝑉 = (6𝑤𝑤 − 5), 𝐾𝐾𝐾𝐾 = (2𝑦𝑦 − 2), 3

𝑅𝑅𝑅𝑅 = (8𝑤𝑤 + 8), and 𝑇𝑇𝑇𝑇 = (3𝑥𝑥 − 4).

1.

∆𝑆𝑆𝑆𝑆𝑆𝑆 is shown, where ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 is a midsegment. 𝑆𝑆𝑆𝑆 = 18𝑥𝑥 + 22 and 𝐷𝐷𝐷𝐷 = 11𝑥𝑥 + 7.

Priscilla was finding the value of 𝑥𝑥. Her work is shown.

1.

Find the value of each variable.

𝑥𝑥 =

2.

𝑦𝑦 =

𝑤𝑤 =

Complete the table.

𝑇𝑇𝑇𝑇 =

𝑉𝑉𝑉𝑉 = 𝑉𝑉𝑉𝑉 = 224

Priscilla’s Work 18𝑥𝑥 + 22 = 11𝑥𝑥 + 7 7𝑥𝑥 = −15 15 𝑥𝑥 = − 7

a.

Write a note to Priscilla explaining her mistake.

b.

Find the value of 𝑥𝑥.

𝐾𝐾𝐾𝐾 =

𝐾𝐾𝐾𝐾 = 𝑅𝑅𝑅𝑅 =

Unit 11: Justifying Relationships in Polygons With Circles

Unit 11, Lesson 9: Solving Problems Using Triangle Relationships

11.9.1 Warm-Up: Would You Rather?

1.

Would you rather use a coupon worth … $20 off your entire purchase

Benchmark MA.912.GR.1.3 Prove relationships and theorems about triangles. Solve mathematical and real-world problems involving postulates, relationships, and theorems of triangles.

2.

OR

25% off your entire purchase?

Explain your choice.

Learning Target 

I can solve mathematical and real-world problems involving postulates, relationships, and theorems of triangles.

Unit 11: Justifying Relationships in Polygons With Circles

225

11.9.2 Activity: Triangle Relationships 1.

For each problem, record your work and answer in the table.

Problem Letter

Work

D

Answer

A E

B

F

C G

226

Unit 11: Justifying Relationships in Polygons With Circles

11.9.3 Wrap-Up: Reflection 1.

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

Circle the color on the stoplight that best represents your understanding of the lessons in this unit.

Red: I do not understand Yellow: I need some more help Green: I got this, I understand

2.

Complete the following statements if you selected red or yellow.

I am still unsure about …

To fill in this gap, I intend to …

Unit 11: Justifying Relationships in Polygons with Circles

227

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

228

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

Unit 11: Justifying Relationships in Polygons with Circles

Unit 12 - Segment Relationships in Circles 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9

Lesson Solving Problems With Lengths of Intersecting Chords Solving Problems With Lengths of Two Secants Solving Problems With Secant and Tangent Segments Solving Problems With Lengths of Two Tangents Pythagorean Theorem and Tangent Lines Lengths of Secants, Tangents, Segments, and Chords in Circles Deriving the Equation of a Circle Standard Form of a Circle Graphing Circles

Unit 12: Segment Relationships in Circles

Page Number 231 237 242 246 251 257 260 266 272

229

The following B.E.S.T. Standards will be covered in this unit: MA.912.GR.6.1 Solve mathematical and real-world problems involving the length of a secant, tangent, segment, or chord in a given circle.

• Clarification 1: Problems include relationships between two chords; two secants; a secant and a tangent; and the length of the tangent from a point to a circle.

MA.912.GR.7.2 Given a mathematical or real-world context, derive and create the equation of a circle using key features.

• Clarification 1: Instruction includes using the Pythagorean Theorem and completing the square.

• Clarification 2: Within the Geometry course, key features are limited to radius, diameter and the center.

MA.912.GR.7.3 Graph and solve mathematical and real-word problems that are modeled with an equation of a circle.

Determine and interpret key features in terms of the context.

• Clarification 1: Key features are limited to domain, range, eccentricity, center and radius.

• Clarification 2: Instruction includes representing the domain and range with inequality notation, interval notation or set-builder notation.

• Clarification 3: Within the Geometry course, notations for domain and range are limited to inequality and set builder.

230

Unit 12: Segment Relationships in Circles

Unit 12, Lesson 1: Solving Problems With Lengths of Intersecting Chords Benchmark

MA.912.GR.6.1 Solve mathematical and real-world problems involving the length of a secant, tangent, segment, or chord in a given circle.

Learning Target 

I can solve mathematical and real-world problems involving lengths formed by the intersection of two chords.

12.1.1 Warm-Up: Facing Challenges “A fixed mindset about ability leads to pessimistic explanations of adversity, and that, in turn, leads to both giving up on challenges and avoiding them in the first place. In contrast, a growth mindset leads to optimistic ways of explaining adversity, and that, in turn, leads to perseverance and seeking out new challenges that will ultimately make you even stronger.” ― Angela Duckworth, Grit: The Power of Passion and Perseverance It took J.K. Rowling seven years to write her first book, Harry Potter and the Sorcerer’s Stone. All twelve major publishing companies rejected her book. Today she is an award-winning author, but at one point in her life it appeared she was a failure.

“It is impossible to live without failing at something, unless you live so cautiously that you might as well not have lived at all — in which case, you fail by default.” — J.K. Rowling

1.

What has challenged you the most in Geometry?

2.

Write an optimistic statement about the challenge.

Unit 12: Segment Relationships in Circles

231

b.

12.1.2 Exploration: Chords 1.

Circles 𝑅𝑅, 𝐺𝐺, and 𝑀𝑀 are shown. Each circle has a chord.

Use patty paper to copy the circles. Use the patty paper copy to construct a perpendicular bisector for each chord by folding.

Summary of Partner’s Conjecture

c. a.

232

Write a conjecture about the perpendicular bisector of a chord of a circle.

Share your conjecture with your partner. Listen to your partner’s conjecture and write a summary.

Have your partner initial that it is summarized correctly.

Partner’s Initials

Review your conjecture and make any revisions, if needed.

Unit 12: Segment Relationships in Circles

2.

Circle 𝐻𝐻 is shown, ̅​̅​̅​̅ are ̅​̅​̅​̅ and 𝐶𝐶𝐶𝐶 where 𝐾𝐾𝐾𝐾 radii of the circle. ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 ̅​̅​̅​̅ are chords of and 𝑃𝑃𝑃𝑃 the circle equidistant from the center 𝐻𝐻.

Therefore, ̅​̅​̅​̅ 𝐿𝐿𝐿𝐿 ≅ ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 . a.

̅​̅​̅​̅ ≅_____ 𝑃𝑃𝑃𝑃

c.

d.

Since ∆𝐷𝐷𝐷𝐷𝐷𝐷 ≅ ∆𝑆𝑆𝑆𝑆𝑆𝑆, what is true about ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 and ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆?

f.

What is true about chords ̅​̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 and ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆?

Using your conjecture about perpendicular bisectors, complete each congruence statement. ̅​̅​̅​̅ ≅_____ 𝐷𝐷𝐷𝐷

b.

e.

Draw a line segment from point 𝐷𝐷 to point 𝐻𝐻 and a line segment from point 𝑆𝑆 to point 𝐻𝐻.

Justify why ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 and ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 are congruent.

̅​̅​̅​̅ ≅ 𝐻𝐻𝐻𝐻 ̅​̅​̅​̅ and ̅​̅​̅​̅ ̅​̅​̅​̅ , what triangle Given that 𝐿𝐿𝐿𝐿 𝑆𝑆𝑆𝑆 ≅ 𝐷𝐷𝐷𝐷 congruence theorem justifies that ∆𝐷𝐷𝐷𝐷𝐷𝐷 and ∆𝑆𝑆𝑆𝑆𝑆𝑆?

Unit 12: Segment Relationships in Circles

233

3.

Circle 𝑌𝑌 is shown, where points 𝐶𝐶, 𝑃𝑃, 𝐾𝐾, and 𝐵𝐵 are on circle 𝑌𝑌. Chords 𝑃𝑃𝑃𝑃 and 𝐾𝐾𝐾𝐾 intersect at point 𝑀𝑀.

12.1.3 Bring It Together: Chord Relationships 1.

Observations of the Blue Chord Measures

Circle 𝑇𝑇 is shown with chord 𝑁𝑁𝑁𝑁 and secant 𝑋𝑋𝑋𝑋, where ⃡𝑋𝑋𝑋𝑋 bisects ̅​̅​̅​̅​̅ 𝑁𝑁𝑁𝑁 and ̅​̅​̅​̅ 𝑋𝑋𝑋𝑋 is a diameter of the circle.

Observations of the Green Chord Measures What information is known about circle 𝑇𝑇, chord 𝑁𝑁𝑁𝑁, ⃡ and 𝑁𝑁𝑁𝑁 ̅​̅​̅​̅​̅ ? and the angles formed by 𝑋𝑋𝑋𝑋

Commonalities in the Blue and Green Chords

What do you wonder about the relationship between the blue chord and the green chord?

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

234

Unit 12: Segment Relationships in Circles

2.

Circle 𝑌𝑌 is shown with chords 𝑋𝑋𝑋𝑋 and 𝑃𝑃𝑃𝑃, where ̅​̅​̅​̅ ≅ 𝑊𝑊𝑊𝑊 ̅​̅​̅​̅​̅. 𝐶𝐶𝐶𝐶

3.

Circle 𝑅𝑅 is shown with chords 𝐻𝐻𝐻𝐻 and 𝐿𝐿𝐿𝐿 that intersect at point 𝐹𝐹.

What information is known about circle 𝑅𝑅, chords 𝐻𝐻𝐻𝐻 and 𝐿𝐿𝐿𝐿, and the angles formed?

What information is known about circle 𝑌𝑌, chords 𝑋𝑋𝑋𝑋 and 𝑃𝑃𝑃𝑃, and the angles formed by ⃡𝐶𝐶𝐶𝐶 ?

If two chords are equidistant from the center of a circle, then the chords are congruent.

When two chords intersect each other inside a circle, the products of their segments are equal.

Unit 12: Segment Relationships in Circles

235

12.1.4 Guided Practice: Solving Problems Involving Intersecting Chords 1.

2.

In circle 𝑍𝑍, chords 𝑃𝑃𝑃𝑃 and 𝐹𝐹𝐹𝐹 intersect at point 𝐷𝐷. Points 𝑃𝑃, 𝐹𝐹, 𝑀𝑀, and 𝑉𝑉 lie on circle 𝑍𝑍. Given 𝐹𝐹𝐹𝐹 = 8, 𝐷𝐷𝐷𝐷 = 12, 𝑉𝑉𝑉𝑉 = 24, and 𝐷𝐷𝐷𝐷 = 3𝑥𝑥 + 1, solve for 𝑥𝑥.

3.

Circle 𝐶𝐶 is shown, where points 𝑇𝑇, 𝑅𝑅, 𝑊𝑊, and 𝑀𝑀 are on the circle. The diameter, 𝑊𝑊𝑊𝑊, is a perpendicular bisector of chord 𝑀𝑀𝑀𝑀. If 𝑀𝑀𝑀𝑀 = (𝑦𝑦 + 4) and 𝐺𝐺𝐺𝐺 = (2𝑦𝑦 + 1), what is the length of the chord 𝑀𝑀𝑀𝑀?

Circle 𝑅𝑅 with points 𝐶𝐶, 𝐾𝐾, 𝐵𝐵, and 𝑃𝑃 on the circle are shown with chords 𝐶𝐶𝐶𝐶 and 𝐾𝐾𝐾𝐾 equidistant from the center. Find the value of 𝑥𝑥, if 𝐶𝐶𝐶𝐶 = (6𝑥𝑥 − 7), 𝐾𝐾𝐾𝐾 = 35, 𝑍𝑍𝑍𝑍 = 𝐻𝐻𝐻𝐻, and 𝑚𝑚∠𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑚𝑚∠𝐾𝐾𝐾𝐾𝐾𝐾 = 90°.

236

Unit 12: Segment Relationships in Circles

12.1.5 Wrap-Up: Reflection

Draw an X on each line to indicate how you feel about solving problems involving chords.

1.

I can use the proportional relationship between the measures of segments formed by intersecting chords to solve problems.

I need help.

I can’t get started.

2.

I understand this and I feel confident.

I can demonstrate that chords are congruent, if the two chords are equidistant from the center of a circle.

I need help.

I can’t get started.

3.

I’m getting there, but I need more practice.

I’m getting there, but I need more practice.

I understand this and I feel confident.

I can show that if a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord.

I need help.

I can’t get started.

I’m getting there, but I need more practice.

I understand this and I feel confident.

Unit 12, Lesson 2: Solving Problems With Lengths of Two Secants Benchmark

MA.912.GR.6.1 Solve mathematical and real-world problems involving the length of a secant, tangent, segment, or chord in a given circle.

Learning Target 

I can solve mathematical and real-world problems that involve the lengths formed by the intersection of two secants.

Unit 12: Segment Relationships in Circles

237

12.2.2 Exploration: Intersecting Secant Lengths

12.2.1 Warm-Up: Would You Rather?

1.

Would you rather have …

A cube of gold, 25 millimeters(mm) on each side …

2.

Explain your choice.

OR

Two cubes of gold, one is 24 mm per side, one is 1 mm per side …

1.

A secant is a line that passes through two points on a circle. In the interior of the circle, a secant line forms a chord.

Circle 𝑅𝑅 is shown with points 𝐴𝐴, 𝑁𝑁, 𝐷𝐷, and 𝐶𝐶 on the circle. Secants ̅​̅​̅​̅ 𝐴𝐴𝐴𝐴 and ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 intersect at point 𝑇𝑇 outside the circle. 𝐴𝐴𝐴𝐴 = 6, 𝐶𝐶𝐶𝐶 = 9, 𝑁𝑁𝑁𝑁 = 8, and 𝐶𝐶𝐶𝐶 = 7.

a.

Complete the table.

Product 𝐴𝐴𝐴𝐴 ∙ 𝑁𝑁𝑁𝑁

𝐴𝐴𝐴𝐴 ∙ 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 ∙ 𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 ∙ 𝐶𝐶𝐶𝐶

𝐷𝐷𝐷𝐷 ∙ 𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷 ∙ 𝐶𝐶𝐶𝐶

238

Unit 12: Segment Relationships in Circles

Work

Answer

b.

Discuss the following with your partner.

•

12.2.3 Guided Practice: Solving Problems Involving Intersecting Secant Lengths

What do you notice about your answers?

1.

•

c.

What do you notice about the secant segments for the rows that have the same answer?

With your partner, use the diagram of Circle 𝑇𝑇 to write a formula that can be used to find the segment lengths of intersecting secants.

c

Circle 𝐴𝐴 with secants ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 and ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 is shown. ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 and ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 intersect at point 𝐷𝐷 outside the circle so that 𝑆𝑆𝑆𝑆 = 5, 𝐷𝐷𝐷𝐷 = 3, 𝑇𝑇𝑇𝑇 = 𝑥𝑥, and 𝐷𝐷𝐷𝐷 = 4. Solve for 𝑥𝑥.

If two secants intersect outside a circle, then the product of the lengths of one secant segment and its external secant segment equals the product of the lengths of the other secant segment and its external secant segment.

Unit 12: Segment Relationships in Circles

239

2.

Circle 𝑀𝑀 is shown, where points 𝑇𝑇, 𝐹𝐹, 𝑁𝑁, and 𝐶𝐶 are on ̅​̅​̅​̅ intersect at point 𝐷𝐷 outside circle 𝑀𝑀. Secants ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 and 𝐶𝐶𝐶𝐶 the circle such that 𝑇𝑇𝑇𝑇 = 5𝑥𝑥, 𝐹𝐹𝐹𝐹 = 4𝑥𝑥, 𝐶𝐶𝐶𝐶 = 7, and 𝑁𝑁𝑁𝑁 = 9. Determine the value of 𝑥𝑥.

3.

240

Leon Sinks Geologic Area is located in Apalachicola National Forest. The area has 18 sinkholes, many of which create a cave system. A park ranger mapped a circular sinkhole. His diagram is ̅​̅​̅​̅ and 𝑉𝑉𝑉𝑉 ̅​̅​̅​̅​̅ such that shown, where circle 𝑆𝑆 has secants 𝑉𝑉𝑉𝑉 points 𝐵𝐵 and 𝐺𝐺 are on circle 𝑆𝑆 and point 𝑉𝑉 is outside the circle. 𝐾𝐾𝐾𝐾 = (4𝑥𝑥 + 4.25) meters (m), 𝑉𝑉𝑉𝑉 = 12.75m, 𝐺𝐺𝐺𝐺 = 3𝑥𝑥 + 5.5 m, and 𝐺𝐺𝐺𝐺 = 14 m. Find the distance, in meters, from point 𝐾𝐾 to point 𝐵𝐵.

Unit 12: Segment Relationships in Circles

12.2.4 Your Turn!

1.

12.2.5 Wrap-Up: Error Analysis

Circle 𝐸𝐸 has points 𝑅𝑅, 𝐹𝐹, 𝐺𝐺, and 𝑀𝑀 on the circle. Secants ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 and ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 intersect at point 𝑍𝑍 outside the circle. Point 𝐺𝐺 is on ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀, and point 𝐹𝐹 is on ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅. 𝑀𝑀𝑀𝑀 = 10, 𝐺𝐺𝐺𝐺 = 8, 𝑅𝑅𝑅𝑅 = (2𝑥𝑥 − 17), and 𝐹𝐹𝐹𝐹 = 9. Solve for 𝑥𝑥.

1.

Donald was finding the value of 𝑥𝑥 in circle 𝑇𝑇, where 𝑌𝑌, 𝑃𝑃, 𝑊𝑊, and 𝐻𝐻 are points on the circle. He knows that ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 and ̅​̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 are secants of the circle, where 𝐶𝐶𝐶𝐶 = 4, 𝑌𝑌𝑌𝑌 = (𝑥𝑥 − 3), 𝐶𝐶𝐶𝐶 = 5, and 𝐻𝐻𝐻𝐻 = (𝑥𝑥 − 6). His work is shown.

Donald’s Work 4(𝑥𝑥 − 3) = 5(𝑥𝑥 − 6) 4𝑥𝑥 − 12 = 5𝑥𝑥 − 30 𝑥𝑥 = 18

2.

Tammie was calculating the distance between her favorite places to visit in Epcot. Circle 𝐸𝐸 represents the World Showcase where 𝑅𝑅, 𝑇𝑇, 𝑀𝑀, and 𝐵𝐵 are on the circle and point 𝑉𝑉 is outside the circle at Spaceship Earth.

Tammie knows that 𝑅𝑅𝑅𝑅 = 975 feet (ft), 𝐵𝐵𝐵𝐵 = 1425 ft, and ̅​̅​̅​̅​̅, in feet? 𝑉𝑉𝑉𝑉 = 1250 ft. What is the length of 𝑀𝑀𝑀𝑀

Write a note to Donald explaining the mistake he made when solving for 𝑥𝑥.

Unit 12: Segment Relationships in Circles

241

Unit 12, Lesson 3: Solving Problems With Secant and Tangent Segments

Benchmark MA.912.GR.6.1 Solve mathematical and real-world problems involving the length of a secant, tangent, segment, or chord in a given circle.

Learning Target 

I can solve mathematical and real-world problems that involve lengths formed by the intersection of a secant and a tangent segment.

242

12.3.1 Warm-Up: Notice and Wonder 1.

The Seminole Native American tribe is known for their novel and colorful dress consisting of patchwork. The decorative technique of patchwork was developed by Seminole women before 1920.

Patchwork was rapidly adopted to further embellish the already colorful clothing. As time went on, the designs became more intricate as the seamstresses became more skilled. a.

What do you notice about the patchwork?

b.

What do you wonder?

Unit 12: Segment Relationships in Circles

12.3.2 Exploration: Lengths of Intersecting Secant and Tangent Segments

b.

Use the diagrams of circle 𝑅𝑅 to write a formula that can be used to find the segment lengths of the intersecting secant and tangent lines.

Work with your partner to complete the following.

1.

Circle 𝐻𝐻 has points 𝑋𝑋, 𝑅𝑅, and 𝑊𝑊 on the circle. Secant ̅​̅​̅​̅​̅ and tangent 𝑋𝑋𝑋𝑋 ̅​̅​̅​̅ intersect at point 𝐷𝐷 outside the 𝑊𝑊𝑊𝑊 circle. Given that ∆𝐷𝐷𝐷𝐷𝐷𝐷~∆𝐷𝐷𝐷𝐷𝐷𝐷, 𝑋𝑋𝑋𝑋 = 15, and 𝑅𝑅𝑅𝑅 = 9.

c c

a.

Use the similiar triangles to write a proportional relationship that can be used to determine the length of ̅​̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 .

If a secant and tangent segment are drawn to a circle from the same external point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.

Unit 12: Segment Relationships in Circles

243

12.3.3 Guided Practice: Solving Problems Involving Intersecting Secant and Tangent Segments 1.

2.

Circle 𝑀𝑀 has points 𝐶𝐶, 𝑉𝑉, and 𝑃𝑃 on the circle. Secant ̅​̅​̅ 𝐶𝐶𝐶𝐶̅ and tangent ̅​̅​̅​̅ 𝑉𝑉𝑉𝑉 intersect at point 𝑆𝑆 outside the circle. 𝐶𝐶𝐶𝐶 = 4 and 𝑆𝑆𝑆𝑆 = 3. What is the length of ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 ? Round to the nearest thousandth.

3.

A drone flew over the Laguna Garzon Bridge in Uruguay and took pictures at three points on the bridge: 𝑁𝑁, 𝑅𝑅, and 𝐻𝐻. Tangent ̅​̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 and secant ̅​̅​̅​̅​̅ 𝑁𝑁𝑁𝑁 intersect at point 𝑊𝑊 off the bridge. If 𝐻𝐻𝐻𝐻 = 280 feet (ft) and 𝑅𝑅𝑅𝑅 = 160 ft, what is the length of ̅​̅​̅​̅​̅ 𝑁𝑁𝑁𝑁 ?

Circle 𝐷𝐷 is shown with points 𝑍𝑍, 𝐴𝐴, and 𝑀𝑀 on the circle.

Secant ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 and tangent ̅​̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 intersect at point 𝐶𝐶 outside the circle. 𝐴𝐴𝐴𝐴 = 16, 𝐴𝐴𝐴𝐴 = (𝑥𝑥 − 3), and 𝑀𝑀𝑀𝑀 = (𝑥𝑥 + 3). What ̅​̅​̅​̅​̅? is the length of 𝐶𝐶𝐶𝐶

244

Unit 12: Segment Relationships in Circles

2.

12.3.4 Your Turn!

1.

Circle 𝑍𝑍 is shown with points 𝐶𝐶, 𝑁𝑁, and 𝐵𝐵 on the circle. ̅​̅​̅​̅ and tangent ̅​̅​̅​̅ Secant 𝐶𝐶𝐶𝐶 𝐵𝐵𝐵𝐵 intersect at point 𝐻𝐻 outside the circle. 𝐶𝐶𝐶𝐶 = (7𝑥𝑥 − 6), 𝑁𝑁𝑁𝑁 = 12, and 𝐻𝐻𝐻𝐻 = 18. Find the value of 𝑥𝑥.

Isabella and Gregory were working on finding the length of ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵. Circle 𝑌𝑌 is given, where 𝐵𝐵, 𝑉𝑉, and 𝐶𝐶 are points on the circle. Secant ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 and tangent ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵 intersect at point 𝑃𝑃 outside the circle. 𝐶𝐶𝐶𝐶 = 12 and 𝑉𝑉𝑉𝑉 = 4.

Isabella’s and Gregory’s work to find the length of ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵 is shown.

Isabella’s Work

Gregory’s Work

𝑥𝑥 2 = 4(4 + 12) 𝑥𝑥 2 = 4(16) 𝑥𝑥 2 = 64 𝑥𝑥 = 8

𝑥𝑥 2 = 12(4 + 12) 𝑥𝑥 2 = 12(16) 𝑥𝑥 2 = 192 𝑥𝑥 = 13.9

a.

Whose work do you agree with?

b.

Write a note to the person whose work is incorrect explaining their mistake.

Unit 12: Segment Relationships in Circles

245

12.3.5 Wrap-Up: Reflection 1.

Choose the emoji that reflects your understanding of solving problems that involve lengths formed by the intersection of a secant and a tangent segment.

Unit 12, Lesson 4: Solving Problems With Lengths of Two Tangents Benchmark

MA.912.GR.6.1 Solve mathematical and real-world problems involving the length of a secant, tangent, segment, or chord in a given circle.

Learning Target



I can solve mathematical and real-world problems that involve lengths formed by the intersection of two tangents.

246

Unit 12: Segment Relationships in Circles

12.4.1 Warm-Up: Bell Work 1.

12.4.2 Exploration: Intersecting Tangent Lines

Circle 𝑇𝑇 has points 𝑋𝑋, 𝑁𝑁, and 𝐵𝐵 on the circle. Secant ̅​̅​̅​̅ 𝑋𝑋𝑋𝑋 ̅​̅​̅​̅ and tangent 𝐵𝐵𝐵𝐵 intersect at point 𝑅𝑅 outside the circle.

Given that 𝑋𝑋𝑋𝑋 = 16 and 𝑁𝑁𝑁𝑁 = 9, find the length of ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵.

1.

⃡ ⃡ and radii 𝑃𝑃𝑃𝑃 Circle 𝑃𝑃 is shown with tangents 𝑀𝑀𝑀𝑀 and 𝑀𝑀𝑀𝑀 and 𝑃𝑃𝑃𝑃.

a. b.

Draw a dotted line from point 𝑃𝑃 to point 𝑀𝑀.

Discuss with your partner the relationships that can ⃡ ⃡ are tangents, be determined given that 𝑀𝑀𝑀𝑀 and 𝑀𝑀𝑀𝑀 ̅​̅​̅​̅ ̅​̅​̅​̅ and 𝑃𝑃𝑃𝑃 and 𝑃𝑃𝑃𝑃 are radii.

Unit 12: Segment Relationships in Circles

247

c.

d.

Discuss with your partner why HL congruence can be used to show that ∆𝑀𝑀𝑀𝑀𝑀𝑀 and ∆𝑀𝑀𝑀𝑀𝑀𝑀 are congruent.

12.4.3 Guided Practice: Solving Problems Involving Intersecting Tangent Lines 1.

̅​̅​̅​̅​̅ Circle 𝐸𝐸 has points 𝑊𝑊 and 𝐾𝐾 on the circle. Tangents 𝑊𝑊𝑊𝑊 ̅​̅​̅​̅ and 𝐾𝐾𝐾𝐾 intersect at point 𝐶𝐶 outside the circle. If 𝑊𝑊𝑊𝑊 = 𝑥𝑥 + 15 and 𝐾𝐾𝐾𝐾 = 2𝑥𝑥 − 5, find the length of ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 .

̅​̅​̅​̅​̅ and 𝑀𝑀𝑀𝑀 ̅​̅​̅​̅​̅?

Since ∆𝑀𝑀𝑀𝑀𝑀𝑀 ≅ ∆𝑀𝑀𝑀𝑀𝑀𝑀, what is true about 𝑀𝑀𝑀𝑀

Tangent segments to a circle from the same point are congruent.

248

Unit 12: Segment Relationships in Circles

2.

Circle 𝑁𝑁 has points 𝐶𝐶, 𝐾𝐾, and 𝑇𝑇 on the circle. Circle 𝐸𝐸 has points 𝑉𝑉, 𝐻𝐻, and 𝐷𝐷 on the circle. Tangents ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 and ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 intersect outside the circle at point 𝐺𝐺. Tangents ̅​̅​̅​̅​̅ 𝑉𝑉𝑉𝑉 ̅​̅​̅​̅​̅ intersect outside the circle at point 𝑊𝑊. and 𝑊𝑊𝑊𝑊 If 𝑇𝑇𝑇𝑇 = (7𝑥𝑥 − 36), 𝐾𝐾𝐾𝐾 = (5𝑥𝑥 − 14), 𝑉𝑉𝑉𝑉 = (8𝑦𝑦 + 13), and 𝑊𝑊𝑊𝑊 = 𝐾𝐾𝐾𝐾, what is the value of 𝑦𝑦?

3.

The Eiffel Tower in Paris, France stands at 324 meters tall. The tower has four semicircular arches at its base.

Circle 𝑇𝑇 represents one of the four arches at the base of the tower, where points 𝑆𝑆 and 𝐴𝐴 lie on the circle. If 𝑆𝑆𝑆𝑆 = (17𝑥𝑥 + 27) meters (m) and 𝐸𝐸𝐸𝐸 = (20𝑥𝑥 − 15) m, what ̅​̅​̅​̅ in meters? is the length of 𝐸𝐸𝐸𝐸

Unit 12: Segment Relationships in Circles

249

2.

12.4.4 Your Turn!

1.

̅​̅​̅​̅ Circle 𝐺𝐺 has points 𝐷𝐷 and 𝐾𝐾 on the circle. Tangents 𝐷𝐷𝐷𝐷 ̅​̅​̅​̅ and 𝐾𝐾𝐾𝐾 intersect at point 𝑌𝑌 outside the circle.

If 𝐷𝐷𝐷𝐷 = 4𝑥𝑥 − 19 and 𝐾𝐾𝐾𝐾 = 15, find the value of 𝑥𝑥.

250

Points 𝑀𝑀 and 𝐵𝐵 are on the circle 𝑉𝑉 and points 𝑅𝑅 and 𝑇𝑇 ̅​̅​̅​̅​̅ and 𝐵𝐵𝐵𝐵 ̅​̅​̅​̅ intersect at point 𝐻𝐻. 𝑀𝑀𝑀𝑀 ̅​̅​̅​̅​̅ are on the circle 𝐷𝐷. 𝑀𝑀𝑀𝑀 is tangent to circle 𝑉𝑉 at point 𝑀𝑀 and tangent to circle 𝐷𝐷 ̅​̅​̅​̅ is tangent to circle 𝑉𝑉 at point 𝐵𝐵 and at point 𝑇𝑇. 𝐵𝐵𝐵𝐵 tangent to circle 𝐷𝐷 at point 𝑅𝑅. If 𝑀𝑀𝑀𝑀 = (4𝑥𝑥 − 8), 𝐵𝐵𝐵𝐵 = (2𝑥𝑥 + 1), 𝑇𝑇𝑇𝑇 = (2𝑥𝑥 − 1), and 𝑅𝑅𝑅𝑅 = (5𝑦𝑦 − 7), what is the value of 𝑦𝑦?

Unit 12: Segment Relationships in Circles

12.4.5 Wrap-Up: Cool Down 1.

Write a short note summarizing today’s lesson to a friend who missed class.

Unit 12, Lesson 5: Pythagorean Theorem and Tangent Lines Benchmark

MA.912.GR.6.1 Solve mathematical and real-world problems involving the length of a secant, tangent, segment, or chord in a given circle.

Learning Target 

I can use the Pythagorean Theorem to solve mathematical and real-world problems involving the length of a tangent segment to a circle.

Unit 12: Segment Relationships in Circles

251

12.5.2 Exploration: Pythagorean Theorem and Tangent Lines

12.5.1 Warm-Up: Ice Problem 1.

An igloo, also known as a snow house or snow hut, is a type of shelter built out of snow.

a.

Work with your partner to complete the following.

1.

Circle 𝑊𝑊 has point 𝐻𝐻 on the circle. ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 and tangent ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 intersect at point 𝐾𝐾 outside the circle.

What is the first question that comes to your mind? a.

b.

252

Discuss with your partner what can be concluded from the given information. Justify your conclusions. Give at least two conclusions.

Write a question that would require someone to use geometry to solve.

Unit 12: Segment Relationships in Circles

b.

Ask a classmate for two conclusions they drew.

Record their conclusions and write your summary of their reason. You are the only person who should write in the first two columns of the table. Have the person initial next to your summary stating that your summary was correct.

Conclusion

My Summary of Their Reason

2.

Circle 𝑅𝑅 is shown with point 𝐷𝐷 outside the circle, where point 𝑁𝑁 is on the circle. 𝐷𝐷𝐷𝐷 = 10, 𝑁𝑁𝑁𝑁 = 8, and 𝐷𝐷𝐷𝐷 = 4√10.

Initials

c.

Discuss with your partner what information is needed to determine the length of ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 .

d.

Explain why any triangle drawn, where the radius and a tangent are the legs, will result in a right triangle.

a.

Use the Pythagorean Theorem to determine if ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 is a tangent to circle 𝑅𝑅.

b.

Complete the statements. ∆𝐷𝐷𝐷𝐷𝐷𝐷 o is o is not

o is o is not

Unit 12: Segment Relationships in Circles

a right triangle. Therefore ̅​̅​̅​̅ 𝐷𝐷𝐷𝐷

a tangent line to circle 𝑅𝑅.

253

12.5.3 Guided Practice: Pythagorean Theorem and Tangent Lines 1.

Circle 𝐵𝐵 has point 𝐻𝐻 on the circle and point 𝑆𝑆 outside the circle. 𝑚𝑚∠𝑆𝑆𝑆𝑆𝑆𝑆 = 90°, 𝑆𝑆𝑆𝑆 = 13, and 𝐵𝐵𝐵𝐵 = 5. a.

What is the length of ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 ?

b.

What is the area of ∆𝑆𝑆𝑆𝑆𝑆𝑆?

254

2.

Circle 𝑃𝑃 has point 𝐻𝐻 on the circle and point 𝑌𝑌 outside the circle, where ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃 and tangent ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 intersect. 𝑃𝑃𝑃𝑃 = 10 and 𝐻𝐻𝐻𝐻 = 17.

What is the length of ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃?

Unit 12: Segment Relationships in Circles

3.

Circle 𝐶𝐶 has points 𝑅𝑅 and 𝑋𝑋 on the circle and point 𝑉𝑉 ̅​̅​̅​̅ and tangent 𝑉𝑉𝑉𝑉 ̅​̅​̅​̅ intersect at outside the circle, where 𝑉𝑉𝑉𝑉 point 𝑉𝑉. 𝑉𝑉𝑉𝑉 = 24 and 𝑉𝑉𝑉𝑉 = 16. a.

What is the length ̅​̅​̅​̅? of 𝐶𝐶𝐶𝐶

b.

̅​̅​̅​̅ ?

What is the length of 𝑉𝑉𝑉𝑉

c.

What is the perimeter of ∆𝑉𝑉𝑉𝑉𝑉𝑉?

12.5.4 Your Turn!

1.

Circle 𝐻𝐻 has point 𝐵𝐵 on the circle and circle 𝐾𝐾 has point 𝐸𝐸 on the circle. ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 and tangent ̅​̅​̅​̅ 𝐸𝐸𝐸𝐸 intersect outside the circles at point 𝑀𝑀. 𝑚𝑚∠𝐻𝐻𝐻𝐻𝐻𝐻 = 𝑚𝑚∠𝑀𝑀𝑀𝑀𝑀𝑀 = 90°, 𝐻𝐻𝐻𝐻 = 6, 𝐵𝐵𝐵𝐵 = 8, and 𝑀𝑀𝑀𝑀 = 8.

a.

Explain why ∆𝐻𝐻𝐻𝐻𝐻𝐻~∆𝐾𝐾𝐾𝐾𝐾𝐾.

b.

What is the length of ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻?

c.

What is the perimeter of ∆𝐾𝐾𝐾𝐾𝐾𝐾?

d.

What is the area of ∆𝐾𝐾𝐾𝐾𝐾𝐾?

Unit 12: Segment Relationships in Circles

255

2.

Grace and Zackary are finding the length of tangent ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 of circle 𝑊𝑊, where 𝐾𝐾 is a point on the circle. ̅​̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 and tangent ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 intersect at point 𝑇𝑇 outside the circle. 𝑇𝑇𝑇𝑇 = 18 and 𝐾𝐾𝐾𝐾 = 7.

12.5.5 Wrap-Up: Ticket Out the Door

1.

Circle 𝑁𝑁 has point 𝑃𝑃 on the circle and point 𝐷𝐷 outside ̅​̅​̅​̅ and tangent 𝐷𝐷𝐷𝐷 ̅​̅​̅​̅. the circle at the intersection of 𝐷𝐷𝐷𝐷 𝑁𝑁𝑁𝑁 = 2 and 𝐷𝐷𝐷𝐷 = 6.

Grace’s and Zackary’s work is shown.

Grace’s Work 72 + 𝑥𝑥 2 = 182 49 + 𝑥𝑥 2 = 324 𝑥𝑥 2 = 275 𝑥𝑥 = 5√11

Zackary’s Work 72 + 182 = 𝑥𝑥 2 49 + 324 = 𝑥𝑥 2 𝑥𝑥 2 = 373 𝑥𝑥 = √373

Write a note to the person whose work is incorrect explaining their mistake.

Means I Start To Acquire Knowledge Experience Skills

̅​̅​̅​̅ ?

What is the length of 𝑁𝑁𝑁𝑁

256

Unit 12: Segment Relationships in Circles

Unit 12, Lesson 6: Lengths of Secants, Tangents, Segments, and Chords in Circles

12.6.1 Warm-Up: Critical Thinking 1.

Explain how math could be considered a language used for communication.

2.

Suppose you were taking Geometry in another country.

How do you think the problems would differ?

Benchmark MA.912.GR.6.1 Solve mathematical and real-world problems involving the length of a secant, tangent, segment, or chord in a given circle.

Learning Target 

I can solve mathematical and real-world problems involving the lengths of secants, tangents, segments, or chords of a circle.

Unit 12: Segment Relationships in Circles

257

12.6.2 Activity: Line Segments of Circles Match each image, type of segment, and answer to the description of circles. Record your answers in the table.

Circle

Circle 𝐾𝐾 has points 𝐺𝐺, 𝑁𝑁, 𝐷𝐷, and 𝐸𝐸 on the circle, where 𝐺𝐺𝐺𝐺 = (2𝑥𝑥 + 2), 𝑀𝑀𝑀𝑀 = 14, 𝐸𝐸𝐸𝐸 = 12, and 𝑀𝑀𝑀𝑀 = (2𝑥𝑥 + 5)

Circle 𝐻𝐻 has points 𝐶𝐶, 𝑇𝑇, and 𝐵𝐵 on the circle, where 𝐵𝐵𝐵𝐵 = 15, 𝑅𝑅𝑅𝑅 = 9, and 𝐶𝐶𝐶𝐶 = 𝑥𝑥

Circle 𝑅𝑅 has points 𝑇𝑇, 𝑆𝑆, 𝑉𝑉, and 𝑊𝑊 on the circle, where 𝑇𝑇𝑇𝑇 = (𝑥𝑥 − 8), 𝑆𝑆𝑆𝑆 = 4, 𝑉𝑉𝑉𝑉 = 3, and 𝑉𝑉𝑉𝑉 = (𝑥𝑥 − 5)

258

Work

Cards’ Letter

Image: ____ Segment: ____ Answer: ____

Image: ____ Segment: ____ Answer: ____

Image: ____ Segment: ____ Answer: ____

Circle 𝐹𝐹 has points 𝐸𝐸 and 𝐵𝐵 on the circle, where 𝐸𝐸𝐸𝐸 = (7𝑥𝑥 − 3) and 𝐵𝐵𝐵𝐵 = (5𝑥𝑥 + 1)

Circle 𝑉𝑉 has point 𝐵𝐵 on the circle, where 𝑉𝑉𝑉𝑉 = 4, 𝐵𝐵𝐵𝐵 = 8 and 𝑉𝑉𝑉𝑉 = 𝑥𝑥

Circle 𝐴𝐴 has points 𝐺𝐺, 𝑅𝑅, 𝐻𝐻, and 𝑆𝑆 on the circle, where 𝑀𝑀𝑀𝑀 = 𝑊𝑊𝑊𝑊, 𝐺𝐺𝐺𝐺 = 35, and 𝑆𝑆𝑆𝑆 = (6𝑥𝑥 − 7)

Unit 12: Segment Relationships in Circles

Image: ____ Segment: ____ Answer: ____

Image: ____ Segment: ____ Answer: ____

Image: ____ Segment: ____ Answer: ____

12.6.3 Try It: Solving for Length 1.

When Doug’s pizza was delivered, he realized the pizza was not cut evenly. The pizza, circle 𝑃𝑃, is cut by chords 𝑁𝑁𝑁𝑁 and 𝐺𝐺𝐺𝐺.

2.

A belt guard is used on stationary bikes to cover some of the inner mechanics. Circle 𝑆𝑆 is shown, where points 𝐻𝐻 and 𝑇𝑇 are on the circle. Tangents ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 and ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 intersect outside the circle at point 𝐺𝐺.

If 𝐻𝐻𝐻𝐻 = 15 and 𝑇𝑇𝑇𝑇 = 4𝑦𝑦 − 9, what is the value of 𝑦𝑦?

Given that 𝑁𝑁𝑁𝑁 = (5𝑥𝑥 − 6) inches (in), 𝐶𝐶𝐶𝐶 = 2 in, 𝐺𝐺𝐺𝐺 = 3 in, ̅​̅​̅​̅? and 𝐶𝐶𝐶𝐶 = (2𝑥𝑥) in, what is the length of 𝑁𝑁𝑁𝑁

Unit 12: Segment Relationships in Circles

259

Unit 12, Lesson 7: Deriving the Equation of a Circle

12.6.4 Wrap-Up: Reflection 1.

One thing I struggled with today was …

Benchmark MA.912.GR.7.2 Given a mathematical or real-world context, derive and create the equation of a circle using key features.

2.

One thing I need to practice more is …

Learning Targets   

3.

I can use the Pythagorean Theorem to derive the equation of a circle.

I can determine key features of a circle given the graph or equation.

I can write the equation of a circle using key features.

One thing I understand very well is …

260

Unit 12: Segment Relationships in Circles

12.7.2 Exploration: Deriving the Equation of a Circle

12.7.1 Warm-Up: Roundabouts Roundabouts promote a continuous flow of traffic because drivers do not have to wait for a green light at a roundabout to get through the intersection. A diagram of a roundabout is shown.

1.

1.

What do you wonder about the diagram?

The center of circle 𝐷𝐷 is located at (0,0). Point 𝑊𝑊 lies on circle 𝐷𝐷 at the point (4, 6.25).

Right triangle 𝐷𝐷𝐷𝐷𝐷𝐷 is drawn inside the circle, such that ̅​̅​̅​̅​̅ 𝐷𝐷𝐷𝐷 is a circle’s radius and point 𝐹𝐹, (4,0), lies on radius 𝐷𝐷𝐷𝐷. a.

Find the length of each side.

Line Segment 2.

Length

̅​̅​̅​̅ 𝐷𝐷𝐷𝐷

How many different paths could the red car take?

̅​̅​̅​̅​̅ 𝐹𝐹𝐹𝐹 b.

3.

List at least two things you have studied in Geometry so far that may be helpful to know when constructing a roundabout.

̅​̅​̅​̅ and Use the values you found for the lengths of 𝐷𝐷𝐷𝐷 ̅​̅​̅​̅​̅ 𝐹𝐹𝐹𝐹 to complete the equation that can be used to ̅​̅​̅​̅​̅ by applying the determine the length of 𝐷𝐷𝐷𝐷 Pythagorean theorem.

(𝐷𝐷𝐷𝐷)2 = (

Unit 12: Segment Relationships in Circles

)2 + (

)2

261

2.

The center of circle 𝑀𝑀 is located at (0,0). Point 𝑃𝑃 lies on circle 𝑀𝑀 at the point (𝑥𝑥, 𝑦𝑦). Right triangle 𝑀𝑀𝑀𝑀𝑀𝑀 is drawn inside the circle, such that ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 is a circle’s radius and point 𝐻𝐻, (𝑥𝑥, 0), lies on radius 𝑀𝑀𝑀𝑀.

a.

3.

The center of circle 𝐶𝐶 is located at (0,0). Point 𝑊𝑊 lies on circle 𝐶𝐶 at the point (−𝑥𝑥, 𝑦𝑦). Right triangle 𝑊𝑊𝑊𝑊𝑊𝑊 is drawn inside the circle, such that ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 is a circle’s radius and point 𝐿𝐿, (−𝑥𝑥, 0), lies on radius 𝑁𝑁𝑁𝑁.

Write an expression that represents the length of each side.

Line Segment

Distance

̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

a.

Write an expression that represents the length of each side.

Line Segment b.

262

̅​̅​̅​̅ 𝐿𝐿𝐿𝐿

Use the expressions you found for the lengths of ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 and ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 to complete the equation that can be ̅​̅​̅​̅​̅ by applying the used to determine the length of 𝑀𝑀𝑀𝑀 Pythagorean theorem.

(𝑀𝑀𝑀𝑀)2 = (

)2 + (

)2

Distance

̅​̅​̅​̅​̅ 𝐿𝐿𝐿𝐿 b.

Use the expressions you found for the lengths of ̅​̅​̅​̅ 𝐿𝐿𝐿𝐿 ̅​̅​̅​̅​̅ to complete the equation that can be used and 𝐿𝐿𝐿𝐿 to determine the length of ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 by applying the Pythagorean theorem.

(𝑊𝑊𝑊𝑊)2 = (

Unit 12: Segment Relationships in Circles

)2 + (

)2

c.

Complete the table with the equation you found that can be used to determine the length of the radius of each circle.

Line Segment

Distance

̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 radius of circle 𝑀𝑀

(𝑀𝑀𝑀𝑀)2 = or (𝑟𝑟)2 =

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 radius of circle 𝐶𝐶

(𝑊𝑊𝑊𝑊)2 = or (𝑟𝑟)2 =

In circle 𝑀𝑀 and circle 𝐶𝐶, the location of the point on the circle was in a different location, but the equations that can be used to determine the length of the radius of the circles are the same.

4.

The center of circle 𝑅𝑅 is located at (3,1). Point 𝑉𝑉 lies on circle 𝑅𝑅 at the point (𝑥𝑥, −𝑦𝑦). Right triangle 𝑅𝑅𝑅𝑅𝑅𝑅 is drawn inside the circle, such that ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 is a circle’s radius and point 𝑍𝑍, (𝑥𝑥, 1), lies on radius 𝑅𝑅𝑅𝑅.

a.

Write an expression that represents the length of each side.

Line Segment

Distance

̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 ̅​̅​̅​̅ 𝑍𝑍𝑍𝑍 b.

Complete the equation that can be used to determine the length of the radius of circle 𝑅𝑅.

(𝑟𝑟)2 = (

Unit 12: Segment Relationships in Circles

)2 + (

)2

263

5.

The center of circle 𝐴𝐴 is located at (ℎ, 𝑘𝑘). Point 𝐵𝐵 lies on circle 𝐶𝐶 at the point (𝑥𝑥, 𝑦𝑦). Right triangle 𝐴𝐴𝐴𝐴𝐶𝐶 is drawn inside the circle such that ̅​̅​̅​̅ 𝐴𝐴𝐴𝐴 is a circle’s radius and point 𝐶𝐶, lies on radius 𝐴𝐴𝐴𝐴. 𝐴𝐴𝐴𝐴 = 𝑏𝑏, 𝐵𝐵𝐵𝐵 = 𝑎𝑎, and 𝐴𝐴𝐴𝐴 = 𝑟𝑟.

a.

b.

Use (𝑥𝑥, 𝑦𝑦) and (ℎ, 𝑘𝑘) to determine the ordered pair for point 𝐶𝐶.

c.

Use the equations in the table to substitute for 𝑎𝑎 and 𝑏𝑏 in the equation 𝑎𝑎2 + 𝑏𝑏2 = 𝑟𝑟 2 .

The center-radius form of an equation of a circle with center, (ℎ, 𝑘𝑘), and radius, 𝑟𝑟, is (𝑥𝑥 − ℎ)2 + (𝑦𝑦 − 𝑘𝑘)2 = 𝑟𝑟 2 .

Fill out the table by using the ordered pairs to complete each equation.

Line Segment ̅​̅​̅​̅ 𝐴𝐴𝐴𝐴

̅​̅​̅​̅ 𝐵𝐵𝐵𝐵

264

The Pythagorean theorem can be used to write the equation, 𝑎𝑎2 + 𝑏𝑏2 = 𝑟𝑟 2 .

Distance 𝑏𝑏 =

𝑎𝑎 =

Unit 12: Segment Relationships in Circles

12.7.3 Guided Instruction: Key Features of a Circle

2.

Circle 𝐹𝐹 is shown, where point 𝐿𝐿 is on the circle.

Circles have two key features: the center and the radius.

The key features can be used to write the equation, (𝑥𝑥 − ℎ)2 + (𝑦𝑦 − 𝑘𝑘)2 = 𝑟𝑟 2 , of the circle.

1.

Circle 𝐾𝐾 is shown with center (3, 4). The point (3, 9) is on circle 𝐾𝐾.

a.

Determine the key features of circle 𝐹𝐹.

Key Features

Radius Center a.

Determine the length of circle 𝐾𝐾’s radius. b.

b.

Write the equation that represents circle 𝐹𝐹.

Write the equation that represents circle 𝐾𝐾.

Unit 12: Segment Relationships in Circles

265

Unit 12, Lesson 8: Standard Form of a Circle

12.7.4 Wrap-Up: Ticket Out the Door 1.

Circle 𝑊𝑊 is shown, where point 𝐵𝐵 is on the circle.

Benchmarks MA.912.GR.7.2 Given a mathematical or real-world context, derive and create the equation of a circle using key features.

MA.912.GR.7.3 Graph and solve mathematical and realworld problems that are modeled with an equation of a circle. Determine and interpret key features in terms of context.

Learning Targets

a.

Determine the key features of circle 𝑊𝑊.

Key Features

  

Radius

I can rewrite the equation of a circle from standard form to center-radius form.

I can determine key features of a circle given the equation in standard form.

I can write the domain and range using inequality notation, interval notation, or set-builder notation.

Center b.

266

Write the equation that represents circle 𝑊𝑊.

Unit 12: Segment Relationships in Circles

12.8.1 Warm-Up: Conic Sections Conic sections are the family of curves created when a plane slices through a cone. A circle is best known as a conic section. A diagram of five conic sections is shown.

12.8.2 Collaboration: Standard Form 1.

The center of circle 𝐻𝐻 is located at (−7,5). Point 𝑇𝑇 lies on circle 𝐻𝐻 at the point (−4.25, 5).

According to Greek philosopher Proclus, Menaechmus, a member of Plato’s Academy, discovered conic sections around 350 B.C. Approximately 100 years later, Apollonius, a Greek geometer and astronomer, developed and published the theory of conic sections. His work earned him the title of the Great Geometer.

a.

1.

Use the diagram to explain why when a cone is sliced horizontally the result is a circle.

Write the equation of the circle in center-radius form, (𝑥𝑥 − ℎ)2 + (𝑦𝑦 − 𝑘𝑘)2 = 𝑟𝑟 2 .

Unit 12: Segment Relationships in Circles

267

b.

Expand each term of the center-radius form to write an equivalent expression.

Term

12.8.3 Refresher: Completing the Square

Equivalent Expression The binomial (𝑥𝑥 + 4)2 can be rewritten as 𝑥𝑥 2 + 8𝑥𝑥 + 16, which is a perfect square trinomial. The algebra tiles represent both (𝑥𝑥 + 4)2 and 𝑥𝑥 2 + 8𝑥𝑥 + 16.

2

(𝑥𝑥 + 7)

(𝑦𝑦 − 5)2 (2.75)2

The standard form of an equation of a circle is 𝑎𝑎𝑎𝑎 2 + 𝑏𝑏𝑦𝑦 2 + 𝑐𝑐𝑐𝑐 + 𝑑𝑑𝑑𝑑 + 𝑒𝑒 = 0.

c.

268

Use the equivalent expressions of the center-radius form to rewrite the center-radius form of circle 𝐻𝐻 in standard form.

1.

Discuss with your partner how the diagram shows that 𝑥𝑥 2 + 8𝑥𝑥 + 16 is a perfect square trinomial. Summarize your discussion.

2.

Discuss with your partner how the diagram shows that 𝑥𝑥 2 + 8𝑥𝑥 + 16 = (𝑥𝑥 + 4)2 .

Unit 12: Segment Relationships in Circles

3.

Write the perfect square trinomial and the binomial squared that are represented by the diagram.

Completing the square by adding 1 tiles will change the value of the original trinomial unless the same number of −1 tiles are also added. b.

4.

Completing the Square 𝑥𝑥 2 − 10𝑥𝑥 + 8 Original Expression 2 𝑥𝑥 − 10𝑥𝑥 + 8 + _______ Completing the square number of 1 tiles added 𝑥𝑥 2 − 10𝑥𝑥 + 8 + _______ + _______ Adding the opposite so the expression is number of −1 tiles added equivalent to the original expression (𝑥𝑥 2 − 10𝑥𝑥 + 25) − 17 Rewriting so that there is a perfect square trinomial perfect square trinomial Writing the perfect (𝑥𝑥 − 5)2 − 17 square trinomial as a binomial squared

The trinomial 𝑥𝑥 2 − 10𝑥𝑥 + 8 is shown.

a.

Complete the table.

How many 1 tiles have to be added to complete the square?

Unit 12: Segment Relationships in Circles

269

12.8.4 Guided Instruction: Rewriting the Standard Form of the Equation of a Circle There are several ways to mathematically represent the domain and range.

Written Inequalities Description

All real numbers greater than or equal to −5 and less than or equal to 5

−5 ≤ 𝑥𝑥 ≤ 5

Set-Builder Notation

Interval Notation

{𝑥𝑥: −5 ≤ 𝑥𝑥 ≤ 5}, {𝑥𝑥| − 5 ≤ 𝑥𝑥 ≤ 5}, {𝑥𝑥: 𝑥𝑥 ≥ −5 and 𝑥𝑥 ≤ 5}, or {𝑥𝑥|𝑥𝑥 ≥ −5 and 𝑥𝑥 ≤ 5}

[−5,5]

2.

𝑥𝑥 2 + 𝑦𝑦 2 − 18𝑥𝑥 − 10𝑦𝑦 = −52 Center-Radius Form

Rewrite each standard form equation of a circle to center-radius form of a circle. Then, give the key features of the circle. Write the domain and range using interval notation and set-builder notation.

1.

𝑥𝑥 2 + 𝑦𝑦 2 − 14𝑥𝑥 + 6𝑦𝑦 + 26 = 0 Center-Radius Form

270

Center

Radius

Domain

Range

Unit 12: Segment Relationships in Circles

Center

Diameter

Domain

Range

12.8.5 Guided Practice: Converting Forms of a Circle 1.

Consider the equation of the circle represented by (𝑥𝑥 + 8)2 + (𝑦𝑦 + 11)2 = 142 . a.

b.

2.

Consider the equation of the circle represented by 𝑥𝑥 2 + 𝑦𝑦 2 − 20𝑥𝑥 = −36. a.

Rewrite 𝑥𝑥 2 + 𝑦𝑦 2 − 20𝑥𝑥 = −36 in center-radius form.

b.

Give the key features of 𝑥𝑥 2 + 𝑦𝑦 2 − 20𝑥𝑥 = −36

Rewrite (𝑥𝑥 + 8)2 + (𝑦𝑦 + 11)2 = 142 in standard form.

Give the key features of (𝑥𝑥 + 8)2 + (𝑦𝑦 + 11)2 = 142 Center

Diameter

Domain

Range

Center

Diameter

Domain

Range

Radius

Radius

Unit 12: Segment Relationships in Circles

271

12.8.6 Wrap-Up: Reflection

1.

Circle the icon that best describes your understanding of each concept.

I need help.

I got it.

I could help someone.

Rewriting center-radius form to standard form

Learning Targets 

Completing the square



Rewriting standard form to center-radius form

Interval notation

272

Benchmark

MA.912.GR.7.3 Graph and solve mathematical and realworld problems that are modeled with an equation of a circle. Determine and interpret key features in terms of context.

Key features of a circle

Set-builder notation

Unit 12, Lesson 9: Graphing Circles

I can graph a circle and determine its key features when given an equation of a circle.

I can solve mathematical and real-world problems that are modeled with an equation of a circle.

Unit 12: Segment Relationships in Circles

12.9.1 Warm-Up: Notice and Wonder The dome of the U.S. Capitol Building is 288 feet (ft) in height and 96 ft in diameter. The dome covers the Rotunda, a room with a circular plan. The architect created an orchestration of architecture, sculpture, and painting in the U.S. Capitol Rotunda.

12.9.2 Guided Practice: Graphing Circles 1.

The equation of circle 𝑊𝑊 is shown.

𝑥𝑥 2 + 𝑦𝑦 2 − 4𝑥𝑥 + 10𝑦𝑦 + 13 = 0

a.

Write the equation in center-radius form.

b.

Complete the table.

Center

c.

1.

What do you notice?

2.

What do you wonder?

Radius

Graph the circle.

Unit 12: Segment Relationships in Circles

273

2.

d.

Find the domain of 𝑥𝑥 2 + 𝑦𝑦 2 − 4𝑥𝑥 + 10𝑦𝑦 + 13 = 0. Write it as an inequality.

b.

What does the center represent in the context?

e.

Find the range of 𝑥𝑥 2 + 𝑦𝑦 2 − 4𝑥𝑥 + 10𝑦𝑦 + 13 = 0. Write it as an inequality.

c.

What does the radius represent in the context?

d.

If the service charges $3.25 for delivery for every four miles driven, what is the most that Marcus could be charged for the delivery?

e.

Determine the domain of (𝑥𝑥 + 4)2 + (𝑦𝑦 − 7)2 = 64.

Write the domain in set-builder notation.

f.

Determine the range of (𝑥𝑥 + 4)2 + (𝑦𝑦 − 7)2 = 64. Write the range in set-builder notation.

Marcus wanted to order from a food delivery service and used the equation (𝑥𝑥 + 4)2 + (𝑦𝑦 − 7)2 = 64 of circle 𝑀𝑀 to find food within his radius, in miles. a.

274

Graph circle 𝑀𝑀.

Unit 12: Segment Relationships in Circles

g.

The grid shows Marcus’ location in green. The blue dots are the locations of restaurants that deliver using UberEats. Use the grid to demonstrate how Marcus could use the domain and range to determine which restaurants he can order from.

12.9.3 Your Turn!

1.

The equation of circle 𝑅𝑅 is (𝑥𝑥 + 2)2 + 𝑦𝑦 2 = 36. a.

Complete the table.

Center Radius Domain Range

b.

Graph circle 𝑅𝑅.

Unit 12: Segment Relationships in Circles

______ ≤ 𝑥𝑥 ≤ ______

______ ≤ 𝑦𝑦 ≤ ______

275

2.

Alyona used the equation (𝑥𝑥 − 3)2 + (𝑦𝑦 + 1) = 81 to determine how far she could sit, in meters, from her Xbox and still use her wireless controller. a.

c.

What does the center represent in the context?

d.

What does the radius represent in the context?

e.

The grid shows the location of Alyona’s Xbox. Use the domain and range to show all of the locations at which Alyona can sit.

Complete the table.

Center Radius Domain Range

b.

276

______ ≤ 𝑥𝑥 ≤ ______

______ ≤ 𝑦𝑦 ≤ ______

Graph (𝑥𝑥 − 3)2 + (𝑦𝑦 + 1) = 81

Unit 12: Segment Relationships in Circles

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

12.9.4 Wrap-Up: Error Analysis 1.

Ziquan and Amber both graphed circle 𝑆𝑆, (𝑥𝑥 − 3)2 + (𝑦𝑦 − 5)2 = 4, but their circles looked different.

Ziquan’s Circle

Amber’s Circle

a.

Who correctly graphed circle 𝑆𝑆?

b.

Write a note to the person who graphed circle 𝑆𝑆 incorrectly explaining their mistake.

Unit 12: Segment Relationships in Circles

277

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

278

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

Unit 12: Segment Relationships in Circles

Unit 13 - Shapes on a Coordinate Plane 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14

Lesson

Parallel and Perpendicular Lines Circles on the Coordinate Plane Determining Points on a Circle Partitioning Line Segments Weighted Averages Classifying Triangles Centroids of Triangles Classifying Quadrilaterals – Part 1 Classifying Quadrilaterals – Part 2 Classifying Quadrilaterals – Part 3 What Is the Figure?

Congruence and Similarity Finding Perimeter and Area on the Coordinate Plane Using Coordinate Geometry to Solve Real-World Problems

Unit 13: Shapes on a Coordinate Plane

Page Number 281 288 294 299 306 314 321 327 334 339 345 348 354 360

279

The following B.E.S.T. Standards will be covered in this unit: MA.912.GR.3.1 Determine the weighted average of two or more points on a line.

• Clarification 1: Instruction includes using a number line and determining how changing the weights moves the weighted average of points on the number line.

MA.912.GR.3.2 Given a mathematical or real-world context, use coordinate geometry to classify or justify definitions, properties and theorems involving circles, triangles or quadrilaterals.

• Clarification 1: Instruction includes using the distance or midpoint formulas and knowledge of slope to classify or justify definitions, properties and theorems.

MA.912.GR.3.3 Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles and quadrilaterals.

• Clarification 1: Problems involving lines include the coordinates of a point on a line segment including the midpoint.

• Clarification 2: Problems involving circles include determining points on a given circle and finding tangent lines.

• Clarification 3: Problems involving triangles include median and centroid.

• Clarification 4: Problems involving quadrilaterals include using parallel and perpendicular slope criteria.

MA.912.GR.3.4 Use coordinate geometry to solve mathematical and real-world problems on the coordinate plane involving perimeter or area of polygons.

MA.912.GR.7.2 Given a mathematical or real-world context, derive and create the equation of a circle using key features.

• Clarification 1: Instruction includes using the Pythagorean Theorem and completing the square.

• Clarification 2: Within the Geometry course, key features are limited to the radius, diameter, and the center.

280

Unit 13: Shapes on a Coordinate Plane

13.1.1 Warm-Up: Artificial Intelligence and Machine Learning

Unit 13, Lesson 1: Parallel and Perpendicular Lines Benchmark MA.912.GR.3.2 Given a mathematical context, use coordinate geometry to classify or justify definitions, properties, and theorems involving circles, triangles, or quadrilaterals.

Artificial intelligence enables a machine to simulate human behavior and decisions. Machine learning is when a machine uses artificial intelligence to learn from past data without a programmer having to re-program the machine.

1.

Describe one task you would like a machine to be programmed to do.

Learning Target 

I can use coordinate geometry to classify parallel and perpendicular lines.

Both artificial intelligence and machine learning rely on mistakes to help improve the programming.

2.

Describe a mistake you have made that helped you improve.

Unit 13: Shapes on a Coordinate Plane

281

13.1.2 Collaboration: Parallel and Perpendicular Lines 1.

4.

What similarities do the two equations have?

5.

What differences do the two equations have?

6.

Complete the statement.

Graph the equation 𝑦𝑦 = −2𝑥𝑥 + 5 on the coordinate plane.

The equations 𝑦𝑦 = −2𝑥𝑥 + 5 and 6𝑥𝑥 + 3𝑦𝑦 = −3 resulted in o parallel o perpendicular o vertical

lines. The slopes of the equations are

o equal. o undefined. o opposite reciprocals.

2.

3.

Rewrite the equation 6𝑥𝑥 + 3𝑦𝑦 = −3 in slope-intercept form. 𝑦𝑦 =______________

Graph the equation 6𝑥𝑥 + 3𝑦𝑦 = −3 using slope-intercept form.

282

7.

Rewrite the equation 4𝑦𝑦 − 6𝑥𝑥 = 8 in slope-intercept form. 𝑦𝑦 =______________

Unit 13: Shapes on a Coordinate Plane

8.

Graph the equation 4𝑦𝑦 − 6𝑥𝑥 = 8 using slope-intercept form.

11. Complete the statement.

2

Equation 4𝑦𝑦 − 6𝑥𝑥 = 8 and equation 𝑦𝑦 = − 𝑥𝑥 − 4 form o parallel o perpendicular o vertical

equations are

3

lines; therefore, the slopes of the

o equal. . o undefined. o opposite reciprocals.

12. Complete the table.

Type

Sketch

Slope

Parallel lines

9.

2

Graph the equation 𝑦𝑦 = − 𝑥𝑥 − 4.

3

10. What do you notice about the slopes in the two equations?

Perpendicular lines

Unit 13: Shapes on a Coordinate Plane

283

13. Match the two lines that are parallel and the two lines that are perpendicular. Then, match the equations.

Lines

13.1.3 Guided Instruction: Parallel and Perpendicular Lines

Equations Lines are parallel if the slopes of the lines are the same.

Parallel Perpendicular Line A

Lines are perpendicular if the slopes are opposites and reciprocals.

Line B

1.

Line C

Equation E 2 𝑦𝑦 + 2 = (𝑥𝑥 + 9) 3 Equation G −2𝑥𝑥 + 3𝑦𝑦 = −6

284

Four lines are shown on the coordinate plane.

Line D

Equation F

a.

What do you notice about the four lines?

2𝑥𝑥 + 3𝑦𝑦 = −21 Equation H 3 𝑦𝑦 = 𝑥𝑥 + 5 2

Unit 13: Shapes on a Coordinate Plane

b.

In the graph of the four lines, the four lines create a rectangle. Explain how each of the properties of a rectangle could be shown.

Property

2.

Line 𝑊𝑊 has points (−1,6) and (1,2) on it. Line 𝑇𝑇 has points (−4,3) and (−2, −2) on it.

Explanation

Opposite sides are parallel.

All angles are right angles.

a. b.

Graph line 𝑊𝑊 and line 𝑇𝑇.

Determine if line 𝑊𝑊 and line 𝑇𝑇 are parallel lines, perpendicular lines, or neither. Justify your answer with math.

Unit 13: Shapes on a Coordinate Plane

285

3.

1

13.1.4 Guided Practice: Parallel and Perpendicular Lines

Line 𝑁𝑁 is written as 𝑦𝑦 = 𝑥𝑥 − 8. Line 𝑉𝑉 is written as 6𝑥𝑥 + 2𝑦𝑦 = 4.

3

Determine if line 𝑁𝑁 and line 𝑉𝑉 are parallel lines, perpendicular lines, or neither. Justify your answer with math.

1.

Given that line 𝑀𝑀 is 3𝑥𝑥 − 𝑦𝑦 = 24, determine if the following lines are parallel to line 𝑀𝑀.

Complete the table.

Equation of Line

𝑦𝑦 = 3𝑥𝑥 + 8

𝑦𝑦 = −3𝑥𝑥 − 1

3𝑦𝑦 = 𝑥𝑥 − 4

3𝑥𝑥 − 𝑦𝑦 = 3

286

Unit 13: Shapes on a Coordinate Plane

Work

Is the line parallel to line 𝑴𝑴? o Yes o No

o Yes o No

o Yes o No

o Yes o No

2.

Line 𝑅𝑅 is written as 4𝑥𝑥 + 𝑦𝑦 = 5, line 𝐾𝐾 is written as 12𝑦𝑦 = 4𝑥𝑥 − 12, and line 𝐶𝐶 is written as 4𝑦𝑦 − 12 = 𝑥𝑥.

13.1.5 Wrap-Up: Reflection 1.

Choose the question and sign that best describes your understanding of today’s lesson.

Are you confused?

2.

a. b.

Are you in the school zone?

Are you ready for the highway?

Briefly explain why you feel this way.

Graph lines 𝑅𝑅, 𝐾𝐾, and 𝐶𝐶.

Determine which two lines are perpendicular lines.

Justify your answer with math.

Unit 13: Shapes on a Coordinate Plane

287

Unit 13, Lesson 2: Circles on the Coordinate Plane

13.2.1 Warm-Up: Bell Work 1.

Benchmarks MA.912.GR.3.2 Given a mathematical context, use coordinate geometry to classify or justify definitions, properties, and theorems involving circles, triangles, or quadrilaterals.

MA.912.GR.3.3 Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles, and quadrilaterals.

Which equation represents a line that is perpendicular 1 to the line 𝑦𝑦 = − 3 𝑥𝑥 + 3?

A. B. C. D.

3𝑥𝑥 + 𝑦𝑦 = 4 𝑥𝑥 − 3𝑦𝑦 = 24 𝑦𝑦 − 3𝑥𝑥 = −4 𝑥𝑥 + 3𝑦𝑦 = 3

MA.912.GR.3.4 Use coordinate geometry to solve mathematical and real-world problems on the coordinate plane involving perimeter or area of polygons.

Learning Targets   

I can use coordinate geometry to determine parts of a circle in a mathematical context.

I can draw a circle using given definitions, properties, or theorems in a mathematical context.

I can use coordinate geometry to solve mathematical problems involving perimeter and area of polygons.

288

Unit 13: Shapes on a Coordinate Plane

13.2.2 Exploration: Circles on the Coordinate Plane 1.

Circle 𝐻𝐻 is shown, where point 𝐶𝐶 is on the circle and line 𝑃𝑃𝑃𝑃 is tangent to circle 𝐻𝐻.

2.

a.

Use a straightedge to draw a line that goes through points 𝐶𝐶 and 𝐻𝐻.

b.

Discuss with your partner how the slope of tangent line 𝑃𝑃𝑃𝑃 and the slope of the radius 𝐶𝐶𝐶𝐶 are related.

c.

What property of circles explains this relationship?

d.

Write the equation of line 𝑃𝑃𝑃𝑃.

e.

Write the equation of line 𝐶𝐶𝐶𝐶.

f.

Line 𝑃𝑃𝑃𝑃 and line 𝐶𝐶𝐶𝐶 intersect at point 𝐶𝐶 (7, −1). Use the equations of lines 𝑃𝑃𝑃𝑃 and 𝐶𝐶𝐶𝐶 to confirm this point.

Circle 𝑀𝑀 has a center at (−2,1) and a tangent line with the equation 𝑦𝑦 = 2𝑥𝑥 − 5. a.

Write the equation of the line perpendicular to the tangent line.

b.

Find the point where the tangent line touches circle 𝑀𝑀. Call this point 𝑅𝑅.

c.

What is the radius of circle 𝑀𝑀?

d.

Write the equation of circle 𝑀𝑀.

Unit 13: Shapes on a Coordinate Plane

289

3.

Circle 𝑊𝑊 is shown, with secant lines 𝑅𝑅𝑅𝑅 and 𝑁𝑁𝑁𝑁, tangent line 𝑁𝑁𝑁𝑁, and chord 𝑅𝑅𝑅𝑅.

c.

Complete the table.

Line

Slope

Equation in Point-Slope Form

⃡ 𝑁𝑁𝑁𝑁 ⃡𝑅𝑅𝑅𝑅 ⃡𝑁𝑁𝑁𝑁

a.

e.

Rewrite the equation of ⃡𝑁𝑁𝑁𝑁 in slope-intercept form.

f.

Use the slope-intercept forms of ⃡𝑅𝑅𝑅𝑅 and ⃡𝑁𝑁𝑁𝑁 to find point 𝐹𝐹.

Length

̅​̅​̅​̅ 𝑁𝑁𝑁𝑁 ̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

290

⃡ in slope-intercept form.

Rewrite the equation of 𝑅𝑅𝑅𝑅

Complete the table.

Line Segment

b.

d.

̅​̅​̅​̅ .

Determine the length of 𝑁𝑁𝑁𝑁

Unit 13: Shapes on a Coordinate Plane

g.

Work with your partner to determine if ∆𝐵𝐵𝐵𝐵𝐵𝐵 is isosceles. Show your work as justification.

13.2.3 Try It: Circles on the Coordinate Plane 1.

h.

What could be used to justify that ∆𝑅𝑅𝑅𝑅𝑅𝑅 is a right triangle?

i.

What is the area of ∆𝑅𝑅𝑅𝑅𝑅𝑅?

j.

What is the perimeter of ∆𝑅𝑅𝑅𝑅𝑅𝑅?

Circle 𝐺𝐺 has a center at (9, −2) and a tangent line with the equation 3𝑥𝑥 + 2𝑦𝑦 = 36. a.

At what point does the tangent line touch circle 𝐺𝐺?

b.

Write the equation of circle 𝐺𝐺.

Unit 13: Shapes on a Coordinate Plane

291

2.

Circle 𝑆𝑆 is shown with secant lines 𝐶𝐶𝐶𝐶 and 𝐺𝐺𝐺𝐺, tangent line 𝑉𝑉𝑉𝑉, and chord 𝐶𝐶𝐶𝐶.

b.

What is the perimeter of ∆𝐶𝐶𝐶𝐶𝐶𝐶?

c.

Complete the table.

Line Slope

⃡𝐺𝐺𝐺𝐺

⃡ 𝐶𝐶𝐶𝐶

Equation in Point-Slope Form

Equation in SlopeIntercept Form a.

Complete the table.

Line Segment ̅​̅​̅​̅​̅ 𝑉𝑉𝑉𝑉

d.

Length

Use the slope-intercept form of both lines to find point 𝑇𝑇.

̅​̅​̅​̅ 𝐶𝐶𝐶𝐶

̅​̅​̅​̅​̅ 𝐶𝐶𝐶𝐶 ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 292

Unit 13: Shapes on a Coordinate Plane

3.

Circle 𝑀𝑀 is shown with tangents ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 and ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅.

a.

What is the length of ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅?

b.

The center of the circle is located at (−6, 4). Write the equation of circle 𝑀𝑀.

c.

What type of quadrilateral is 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾? Justify your choice.

d.

What is the distance between point 𝑅𝑅 and point 𝑀𝑀?

13.2.4 Wrap-Up: Growth Mindset 1.

Today, I am still unsure about …

2.

To fill in this gap, I intend to …

3.

One question I still have is …

Unit 13: Shapes on a Coordinate Plane

293

Unit 13, Lesson 3: Determining Points on a Circle Benchmarks MA.912.GR.3.2 Given a mathematical context, use coordinate geometry to classify or justify definitions, properties, and theorems involving circles, triangles, or quadrilaterals.

13.3.1 Warm-Up: Notice and Wonder Castlerigg Stone Circle is one of the earliest stone circles to be found in Britain and is important in terms of megalithic astronomy and geometry. It is composed of 38 free standing stones, some measuring up to 10 feet high and dating back to the Neolithic period, 4000 to 5000 years ago.

MA.912.GR.3.3 Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles, and quadrilaterals.

MA.912.GR.7.2 Given a mathematical or real-world context, derive and create the equation of a circle using key features.

1.

What do you notice?

2.

What do you wonder?

Learning Targets   

I can determine whether a point is on the circumference of a circle, in a circle, or outside a circle.

I can find the center of a circle when given two points.

I can write the equation of a circle given the endpoints of a diameter.

294

Unit 13: Shapes on a Coordinate Plane

13.3.2 Exploration: Points on a Circle 1.

The equation of circle 𝐻𝐻 is (𝑥𝑥 + 4)2 + (𝑦𝑦 − 3)2 = 36. a. Complete the table by substituting the given coordinates for 𝑥𝑥 and 𝑦𝑦 in the equation of circle 𝐻𝐻.

Then, determine the value of the left side of the equation to complete the simplified equation in the last column.

Coordinate

Substitution

d.

Plot points 𝑀𝑀(−8, −2), 𝐵𝐵(−3, −1), and 𝑅𝑅(−4, −3).

e.

What do you notice about the relationship between the location of the points and their simplified equations?

f.

Write a conjecture about the location of a point and the equation of the circle.

Simplified Equation

𝑀𝑀(−8, −2)

_____ = 36

𝑅𝑅(−4, −3)

_____ = 36

𝐵𝐵(−3, −1)

_____ = 36

b.

What do you notice about the simplified equations?

c.

Graph circle 𝐻𝐻, (𝑥𝑥 + 4)2 + (𝑦𝑦 − 3)2 = 36.

Unit 13: Shapes on a Coordinate Plane

295

g.

Ask two classmates for their conjectures. Record their conjectures and write your summary of their conjecture. You are the only person who should write in the first two columns of the table. Have the person initial next to your summary stating that your summary was correct.

Conjecture

My Summary of Their Conjecture

2.

The equation of circle 𝐾𝐾 is (𝑥𝑥 + 6)2 + (𝑦𝑦 + 5)2 = 25 with points 𝑁𝑁(−6,0), 𝑉𝑉(−3, −1), 𝑇𝑇(−1, −5), and 𝑅𝑅(−9, −9) on the circle. a.

What is the length of the diameter of circle 𝐾𝐾?

b.

Complete the table by finding the distance between the given points.

Initials

Points

Work

Distance

𝑁𝑁(−6,0) and 𝑅𝑅(−9, −9)

𝑉𝑉(−3, −1) and 𝑅𝑅(−9, −9) 𝑇𝑇(−1, −5) and 𝑅𝑅(−9, −9)

h.

296

Review your conjecture with your partner. If necessary, make any changes.

c.

What do you notice about the distances between the points?

d.

What can you conclude about points 𝑉𝑉 and 𝑅𝑅?

Unit 13: Shapes on a Coordinate Plane

3.

Circle 𝑊𝑊 has a radius of √29 and points 𝑆𝑆(2, −2), 𝐷𝐷(5,1), and 𝐻𝐻(12, −6). a.

Determine which two points are the endpoints of a diameter of circle 𝑊𝑊. Show your work.

b.

Describe how to find the center of the circle using the two endpoints of the diameter.

c.

What is the center of the circle?

d.

Write the equation for circle 𝑊𝑊.

4.

The points 𝐺𝐺 (3, −1) and 𝐻𝐻 (3, 3) are the endpoints of a diameter of circle 𝑀𝑀. Write the equation of circle 𝑀𝑀.

Unit 13: Shapes on a Coordinate Plane

297

2.

13.3.3 Try It: Points on a Circle 1.

The equation of circle 𝑃𝑃 is (𝑥𝑥 − 8)2 + (𝑦𝑦 − 2)2 = 13.

a.

Determine whether the following points lie inside, outside, or on circle 𝑃𝑃.

Point Work Location (5, −1)

o Inside circle 𝑃𝑃 o Outside circle 𝑃𝑃 o On circle 𝑃𝑃

(6,5)

o Inside circle 𝑃𝑃 o Outside circle 𝑃𝑃 o On circle 𝑃𝑃

(12,2)

o Inside circle 𝑃𝑃 o Outside circle 𝑃𝑃 o On circle 𝑃𝑃

(11,0)

o Inside circle 𝑃𝑃 o Outside circle 𝑃𝑃 o On circle 𝑃𝑃

(9,4)

o Inside circle 𝑃𝑃 o Outside circle 𝑃𝑃 o On circle 𝑃𝑃

298

Circle 𝑇𝑇 and circle 𝑅𝑅 intersect at point 𝐻𝐻.

Equation of circle 𝑇𝑇 is 𝑥𝑥 2 + (𝑦𝑦 − 7)2 = 20 Equation of circle 𝑅𝑅 is (𝑥𝑥 − 7)2 + (𝑦𝑦 − 3)2 = 25

3.

Which coordinate pair is the location of point 𝐻𝐻? o (−4,5) o (7,8) o (2,3) o (4,9)

Circle 𝑆𝑆 has a radius of 10 and points 𝐹𝐹(−6, −6), 𝐶𝐶(8,8), 𝑊𝑊(10,2), and 𝑀𝑀(−8, −4) on the circle. a.

Determine which two of the given points are endpoints of a diameter of circle 𝑆𝑆.

b.

Find the center of circle 𝑆𝑆.

c.

Write the equation of circle 𝑆𝑆.

Unit 13: Shapes on a Coordinate Plane

Unit 13, Lesson 4: Partitioning Line Segments

13.3.4 Wrap-Up: Ticket Out the Door 1.

The equation of circle 𝑊𝑊 is 𝑥𝑥 2 + (𝑦𝑦 + 4)2 = 26.

Select all the points that lie on circle 𝑊𝑊.     

(3,0) (5, −3) (1,1) (−3, −2) (−5, −5)

Benchmark MA.912.GR.3.3 Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles, and quadrilaterals.

Learning Target 

I can use coordinate geometry to partition a line segment.

Unit 13: Shapes on a Coordinate Plane

299

13.4.2 Exploration: Partitioning a Line Segment

13.4.1 Warm-Up: Heroic Person 1.

2.

Who do you consider heroic and inspirational?

Why did you pick this person?

1.

̅​̅​̅​̅ is shown.

Directed line segment 𝑆𝑆𝑆𝑆

a.

How many equal parts is ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆 partitioned into?

b.

̅​̅​̅​̅ is partitioned into a 2: 3 Make a mark where 𝑆𝑆𝑆𝑆 ratio. Call this point 𝑅𝑅.

c.

Complete the statement.

Point 𝑅𝑅 is

d.

2

o3 2

o5 o

3 5

of the distance from 𝑆𝑆 to 𝑃𝑃.

Complete the equation that can be used to find the distance of ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆. 𝑆𝑆𝑆𝑆 = _______𝑆𝑆𝑆𝑆

300

Unit 13: Shapes on a Coordinate Plane

2.

Point 𝑋𝑋 partitions directed line segment 𝐾𝐾𝐾𝐾 in a 3: 4 ratio. Points 𝐾𝐾(−9,1), 𝐹𝐹(−6,1), 𝑋𝑋(−6,4), 𝑅𝑅(−2,8), and 𝑊𝑊(−2,1) are shown on the coordinate grid.

c.

Two students were discussing the directed line segment 𝐾𝐾𝐾𝐾.

Nevaeh Isabella Point 𝑅𝑅 is a dilation of point Point 𝑅𝑅 is a dilation of point 𝑋𝑋 centered at point 𝐾𝐾 using 𝑋𝑋 centered at point 𝐾𝐾 using 3 3 a scale factor of 7. a scale factor of .

4

Work with your partner to explain why neither student has the correct scale factor. Include in your explanation what the correct scale factor should be so that the result of dilating at point 𝐾𝐾 is point 𝑅𝑅.

Use the distances to write the numeric ratio for each segment ratio. Then, choose the type of ratio the segment ratio represents.

Segment Numeric Ratio Type of Ratio Ratio o part:part 𝐾𝐾𝐾𝐾: 𝐹𝐹𝐹𝐹 o part:whole o part:part 𝐾𝐾𝐾𝐾: 𝐾𝐾𝐾𝐾 o part:whole o part:part 𝑋𝑋𝑋𝑋: 𝑅𝑅𝑅𝑅 o part:whole o part:part 𝐾𝐾𝐾𝐾: 𝐾𝐾𝐾𝐾 o part:whole

d.

Discuss with your partner which ratios can be used to justify that ∆𝐾𝐾𝐾𝐾𝐾𝐾~∆𝐾𝐾𝐾𝐾𝐾𝐾. Summarize your discussion.

e.

With your partner, write a conjecture on how point 𝑋𝑋 could be found if only given the endpoints of the directed line segment 𝐾𝐾𝐾𝐾 and the ratio.

a.

b.

What do you notice about the numeric ratios?

Unit 13: Shapes on a Coordinate Plane

301

f.

Ask two other pairs of partners for the conjecture they wrote. Listen to their conjectures and write a summary. You are the only person who should write in the first column of the table. Have the person initial next to your summary stating that your summary was accurate.

Conjecture

302

13.4.3 Guided Instruction: Partitioning a Line Segment 1.

Generalize the conjecture in the Exploration if the given ratio is part:whole.

Initials

Unit 13: Shapes on a Coordinate Plane

2.

Directed line segment 𝑀𝑀𝑀𝑀 is shown with endpoints at point 𝑀𝑀(−8,3) and point 𝐾𝐾(7,13).

c.

Use your equation to find the vertical distance of ̅​̅​̅​̅​̅. 𝑀𝑀𝑀𝑀

d.

Use the horizontal distance and the vertical distance to plot point 𝐵𝐵 on ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀.

e.

What are the coordinates of point 𝐵𝐵?

Directed line segment 𝑀𝑀𝑀𝑀 is partitioned by point 𝐵𝐵 in a ratio of 2: 3. a.

b.

Write an equation that can be used to find the ̅​̅​̅​̅​̅. distance of 𝑀𝑀𝑀𝑀

Use your equation to find the horizontal distance of ̅​̅​̅​̅​̅. 𝑀𝑀𝑀𝑀

Unit 13: Shapes on a Coordinate Plane

303

3.

Directed line segment 𝑋𝑋𝑋𝑋 has endpoints at point 𝑋𝑋(−6,1) and point 𝑌𝑌(4,4).

Directed line segment 𝑋𝑋𝑋𝑋 is partitioned from 𝑋𝑋 to 𝑌𝑌 at point 𝑊𝑊. a.

b.

c.

304

1 2

13.4.4 Try It: Partitioning a Line Segment 1.

Directed line segment 𝑊𝑊𝑊𝑊 has endpoints at 𝑊𝑊(−3,3) and 𝐺𝐺(6, −3).

of the distance

What is the horizontal distance of ̅​̅​̅​̅​̅ 𝑋𝑋𝑋𝑋 ?

What is the vertical distance of ̅​̅​̅​̅​̅ 𝑋𝑋𝑋𝑋 ?

What are the coordinates of point 𝑊𝑊?

Directed line segment 𝑊𝑊𝑊𝑊 is partitioned distance from 𝑊𝑊 to 𝐺𝐺 at point 𝐷𝐷.

2 3

of the

a.

̅​̅​̅​̅​̅?

What is the horizontal distance of 𝑊𝑊𝑊𝑊

b.

̅​̅​̅​̅​̅?

What is the vertical distance of 𝑊𝑊𝑊𝑊

c.

What are the coordinates of point 𝐷𝐷?

Unit 13: Shapes on a Coordinate Plane

2.

Directed line segment 𝐵𝐵𝐵𝐵 has endpoints at point 𝐵𝐵(9, −7) and point 𝑁𝑁(−1, −2).

13.4.5 Wrap-Up: Reflection 1.

Choose the emoji that reflects your understanding of partitioning a line segment.

Directed line segment 𝐵𝐵𝐵𝐵 is partitioned by point 𝐻𝐻 at a 4: 1 ratio.

What are the coordinates of point 𝐻𝐻?

Unit 13: Shapes on a Coordinate Plane

305

Unit 13, Lesson 5: Weighted Averages Benchmark MA.912.GR.3.1 Determine the weighted average of two or more points on a line.

Learning Targets  

I can determine the weighted average of two or more points on a line.

I can explain how changing the weight moves the weighted average of points.

13.5.1 Warm-Up: Between Two Numbers A stock exchange, or stock market, is a system for buying and selling bonds, securities, or stocks. A stock is a share in the ownership of a company. NASDAQ is short for National Association of Securities Dealers Automated Quotations, which is the world's largest electronic stock exchange. Each company that is traded has an abbreviation of one to four letters.

Jaxon’s grandmother bought 100 shares of Netflix and 100 shares of Apple for him when he was born on March 29, 2003. She paid $1.37 per share for Netflix (NASDAQ: NFLX) and paid $0.21 per share for Apple (NASDAQ: AAPL).

On Jaxon’s 18th birthday, Netflix’s stock price was at $513.95 and Apple’s stock price was at $121.39.

1.

How much were the stocks worth on Jaxon’s 18th birthday?

The stock price history of Netflix shows that at one point the stock had increased in worth by 45,000%, and Apple stock had increased in worth by 140,000%.

2.

Write the decimal equivalent of each percentage increase.

Stock Apple Netflix

306

Unit 13: Shapes on a Coordinate Plane

Decimal Equivalent

13.5.2 Exploration: Exploring Weighted Averages 1.

using the steps shown.

3 1 3 1 9 15 24 (point 𝐻𝐻) + (point 𝑀𝑀) = (3) + (15) = + = =6 4 4 4 4 4 4 4

The ratio

Find the midpoint of ̅​̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 and label it 𝐵𝐵.

Assuming that points 𝐻𝐻 and 𝑀𝑀 have the same weight, the diagram represents how points 𝐻𝐻, 𝐵𝐵, and 𝑀𝑀 are in balance.

𝑯𝑯

𝑩𝑩

3

1

ratio, 3, can be used to determine the location of point 𝐾𝐾

A number line is shown. Each grid space is 1 unit.

a.

1

The ratio of 𝐻𝐻𝐻𝐻: 𝐾𝐾𝐾𝐾 is 3: 9, which can be written as . The

𝑴𝑴

Discuss with your partner why point 𝐵𝐵 acts as a point of balance. Summarize your discussion.

c.

Keyshawn told his partner that if the scale is drawn where the scale is centered on point 𝐾𝐾, then point 𝐻𝐻 would be higher than point 𝑀𝑀. Describe or draw what would have to be done to balance the scale when it is centered at point 𝐾𝐾.

3

was used to find

3 4

and

1 4

by noting that the ratio

indicates there are a total of 4 parts. To balance at point 𝐾𝐾, point 𝐻𝐻 would be “heavier” than point 𝑀𝑀, so it is weighted as 3 of the 4 parts with point 𝑀𝑀 weighted as 1 of the 4 parts.

2.

b.

1

Point 𝑇𝑇 has coordinates (2, 4), and point 𝑀𝑀 has coordinates (8, 1).

a.

What is the midpoint of line segment 𝑇𝑇𝑇𝑇?

Unit 13: Shapes on a Coordinate Plane

307

b.

Complete the table.

f.

𝒙𝒙-value

𝒚𝒚-value

1 (______) 2

1 (______) 2

1 (______) 2

𝑻𝑻(𝟐𝟐, 𝟒𝟒) 𝑴𝑴(𝟖𝟖, 𝟏𝟏) Total

1 (______) 2

Discuss with your partner how the table shows the process of finding the midpoint.

d.

Complete the expression that can be used to find the point that partitions the line segment in a 1: 1 ratio. 𝑇𝑇 represents the coordinates of point 𝑇𝑇, and 𝑀𝑀 represents the coordinates of point 𝑀𝑀.

1

e.

𝑇𝑇 +

1

𝑀𝑀

Find the point that partitions directed line segment 𝑇𝑇𝑇𝑇 in a 2: 1 ratio.

2 3

𝒚𝒚-value 1 (______) 3

2 (______) 3

Total

2 (______) 3

What do you notice about the total in the table and the point found in part e that partitions the direct line segment in a 2:1 ratio?

Line segment 𝐾𝐾𝐾𝐾 has endpoints at 𝐾𝐾(−6, 1) and 𝑅𝑅(2, 5). a.

b.

308

1 (______) 3

𝑴𝑴(𝟖𝟖, 𝟏𝟏)

3.

3

𝒙𝒙-value

𝑻𝑻(𝟐𝟐, 𝟒𝟒)

g. c.

1

Complete the table to calculate 𝑇𝑇 + 𝑀𝑀.

Complete the expression for the point that partitions directed line segment 𝐾𝐾𝐾𝐾 in a 3: 1 ratio.

4

𝐾𝐾 +

4

𝑅𝑅

Use the expression to find the point that partitions directed line segment 𝐾𝐾𝐾𝐾 in a 3: 1 ratio.

Unit 13: Shapes on a Coordinate Plane

13.5.3 Guided Instruction: Determining Weighted Averages Weight is a measure of force pushing or pulling on an object due to gravity. This means that finding a weighted average puts more “weight” on one point than another.

Directed line segment 𝑀𝑀𝑀𝑀 is shown, where point 𝑀𝑀 is at (1, −11) and point 𝑃𝑃 is at (5, 7). Point 𝐻𝐻 is the midpoint of ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀, point 𝑊𝑊 is the midpoint of ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻, and point 𝑅𝑅 is the ̅​̅​̅​̅​̅. midpoint of 𝑀𝑀𝑀𝑀

If 𝐵𝐵 is weighted twice as heavily as 𝐴𝐴, then during the calculation, 𝐵𝐵 is to measure two times the value of 𝐴𝐴. In other words, 𝐵𝐵 has two times the “gravitational pull” of 𝐴𝐴.

Think of it in terms of a seesaw. In the picture below, point 𝐵𝐵 is weighted more heavily than point 𝐴𝐴.

Since point 𝐻𝐻 is the midpoint of ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀, the coordinates of point 𝐻𝐻 can be determined by finding the average of points 𝑀𝑀 and 𝑃𝑃.

1.

Discuss with your partner what the girls close to point B would have to do so that the seesaw would balance.

Summarize your discussion.

𝑀𝑀+𝑃𝑃 2

Point 𝑊𝑊 partitions the line segment 𝑀𝑀𝑀𝑀 so that point 𝑃𝑃 weighs three times as much as point 𝑀𝑀. This means that point 𝑊𝑊 is closer to point 𝑃𝑃 because point 𝑃𝑃 is “heavier” than point 𝑀𝑀. The given expression can be used to calculate this weighted average when point 𝑃𝑃 weighs three times as much as point 𝑀𝑀. 𝑀𝑀+𝑃𝑃+𝑃𝑃+𝑃𝑃 4

Unit 13: Shapes on a Coordinate Plane

309

4.

The graph shows the weighted averages.

Write the expression that can be used to find the point ̅​̅​̅​̅​̅ in the given ratio, and find the that partitions 𝑀𝑀𝑀𝑀 endpoint that is the weighted point.

Ratio

Expression

1: 1

2.

Discuss with your partner why the distance from point 𝑊𝑊 to point 𝑃𝑃 is shorter if point 𝑃𝑃 is “heavier” than point 𝑀𝑀. Summarize your discussion.

1: 3

3: 1 3.

Point 𝑅𝑅 partitions the line segment 𝑀𝑀𝑀𝑀 so that point 𝑀𝑀 weighs three times as much as point 𝑃𝑃. Write an expression that shows this weighted average.

A midpoint calculation weighs points on a line segment equally. In a weighted average calculation, points on a line segment can be weighted differently.

310

Unit 13: Shapes on a Coordinate Plane

Point

Weighted Point

5.

A line segment is shown in the table with three different weighted averages. Write the expression that can be used to find each weighted average.

Partition

Graph

6.

Discuss with your partner how changing the weights moves the weighted average on the number line.

Summarize your discussion.

Expression

4: 1

2: 3

1: 5

Unit 13: Shapes on a Coordinate Plane

311

13.5.4 Collaboration: Weighted Averages and Dilations 1.

c.

Find the point that partitions ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 in a 1: 4 ratio.

Label it 𝑆𝑆.

d.

Is 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 a dilation of 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊? Justify your answer.

𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 has vertices at 𝑉𝑉(8,9), 𝑋𝑋(6,3), 𝑊𝑊(2,3), and 𝑌𝑌(4,9).

a.

Find the point that partitions ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 in a 1: 4 ratio.

Label it 𝑅𝑅.

b.

Find the point that partitions ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 in a 1: 4 ratio.

Label it 𝑇𝑇.

312

Unit 13: Shapes on a Coordinate Plane

2.

Triangle 𝑁𝑁𝑁𝑁𝑁𝑁 has vertices at 𝑁𝑁(−2,6), 𝐶𝐶(3,6), and 𝐻𝐻(−2, −3).

Use segment partitioning to dilate ∆𝑁𝑁𝑁𝑁𝑁𝑁 using center 𝐶𝐶 3 and scale factor 4.

13.5.5 Your Turn!

1.

Line segment 𝑀𝑀𝑀𝑀 has endpoints at 𝑀𝑀(−2, 1) and 𝐾𝐾(8, 7). a.

Write an expression for the point that partitions directed line segment 𝑀𝑀𝑀𝑀 in a 2: 3 ratio.

b.

Use the expression to find the point that partitions directed line segment 𝑀𝑀𝑀𝑀 in a 2: 3 ratio.

Unit 13: Shapes on a Coordinate Plane

313

Unit 13, Lesson 6: Classifying Triangles

13.5.6 Wrap-Up: Ticket Out the Door 1.

A number line is shown. Each grid space is 1 unit.

a.

Determine the point on the number line that satisfies the expression shown. Label the point 𝑀𝑀.

2 3 (point 𝑊𝑊) + (point 𝑇𝑇) 5 5

Benchmark MA.912.GR.3.2 Given a mathematical context, use coordinate geometry to classify or justify definitions, properties, and theorems involving circles, triangles, or quadrilaterals.

Learning Targets 

b.

314

What is the ratio 𝑊𝑊𝑊𝑊: 𝑀𝑀𝑀𝑀?



I can use coordinate geometry to classify triangles in a mathematical or real-world context.

I can draw a triangle on a coordinate plane using given definitions, properties, or theorems.

Unit 13: Shapes on a Coordinate Plane

13.6.1 Warm-Up: Latitude and Longitude In ancient times, people used landmarks to identify locations, but travelers were not familiar with local landmarks, making traveling difficult. Hipparchus, a Greek astronomer, was the first to specify location using latitude and longitude as coordinates. Hipparchus applied his knowledge of spherical angles to the problem of denoting locations on the Earth’s surface.

13.6.2 Refresher: Finding Distance Work with your partner to complete the following.

1.

Find the length of ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 using the Pythagorean Theorem.

If necessary, round to the nearest thousandth.

Latitude is the measurement of the globe north to south, and longitude is the measurement of the globe east to west.

The table shows the latitude and longitude of different Florida cities.

City Coordinates City Coordinates 30.2𝑁𝑁, 85.7𝑊𝑊 25.8𝑁𝑁, 80.2𝑊𝑊 Panama City Miami 28.0𝑁𝑁, 82.5𝑊𝑊 Tallahassee 30.46𝑁𝑁, 84.28𝑊𝑊 Tampa 26.6𝑁𝑁, 80.1𝑊𝑊 28.5𝑁𝑁, 81.4𝑊𝑊 Lake Worth Orlando 30.3𝑁𝑁, 81.7𝑊𝑊 Jacksonville Gainesville 29.7𝑁𝑁, 82.3𝑊𝑊 1.

What do you notice? What do you wonder?

Unit 13: Shapes on a Coordinate Plane

315

2.

Use the Pythagorean Theorem to find the length of ̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 .

If necessary, round to the nearest thousandth.

13.6.3 Exploration: Classifying Triangles 1.

3.

∆𝑀𝑀𝑀𝑀𝑀𝑀 has vertices at 𝑀𝑀(−8,8), 𝐾𝐾(−4,2), and 𝐶𝐶(−9,3). a.

Graph ∆𝑀𝑀𝑀𝑀𝑀𝑀 on the coordinate plane.

b.

Complete the table.

Explain how the Pythagorean Theorem can be used to find the distance between two points.

Line Segment ̅​̅​̅​̅​̅ 𝐶𝐶𝐶𝐶

̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾

316

Unit 13: Shapes on a Coordinate Plane

Length of Line Segment

c.

What do you notice about the side lengths of ∆𝑀𝑀𝑀𝑀𝑀𝑀?

b.

Complete the table. If necessary, round to the nearest thousandth.

Line Segment

d.

because ∆𝑀𝑀𝑀𝑀𝑀𝑀 has 2.

̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

̅​̅​̅​̅​̅ 𝑅𝑅𝑅𝑅

Complete the statement. ∆𝑀𝑀𝑀𝑀𝑀𝑀 is classified as a(n)

Length of Line Segment

o equilateral o isosceles o scalene

o no o two o three

triangle

equal side lengths.

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊

c.

What do you notice about the side lengths of ∆𝑁𝑁𝑁𝑁𝑁𝑁?

d.

Complete the statement.

∆𝑁𝑁𝑁𝑁𝑁𝑁 has vertices at 𝑁𝑁(6,5), 𝑊𝑊(3, −1), and 𝑅𝑅(10, −1). a.

Graph ∆𝑁𝑁𝑁𝑁𝑁𝑁 on the coordinate plane.

o equilateral ∆𝑁𝑁𝑁𝑁𝑁𝑁 is classified as a(n) o isosceles o scalene because ∆𝑁𝑁𝑁𝑁𝑁𝑁 has o no o two o three

Unit 13: Shapes on a Coordinate Plane

triangle

equal side lengths.

317

3.

∆𝑀𝑀𝑀𝑀𝑀𝑀 has vertices at 𝑀𝑀(−3, 4), 𝑆𝑆(2.2, 1), and 𝐺𝐺(−3, −2). a.

Complete the table. If necessary, round to the nearest tenth.

Line Segment

4.

Draw a scalene triangle with one vertex at (−3,6) and with a base that is twice the length of its height.

Length of Line Segment

̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 ̅​̅​̅​̅ 𝑆𝑆𝑆𝑆

̅​̅​̅​̅​̅ 𝐺𝐺𝐺𝐺 b.

Complete the statement. ∆𝑀𝑀𝑀𝑀𝑀𝑀 is classified as a(n) because ∆𝑀𝑀𝑀𝑀𝑀𝑀 has

318

o no o two o three

o equilateral o isosceles o scalene

triangle

equal side lengths.

Unit 13: Shapes on a Coordinate Plane

2.

13.6.4 Your Turn!

1.

Given: ∆𝑆𝑆𝑆𝑆𝑆𝑆 with 𝑊𝑊(1,6), 𝑆𝑆(7,6), and 𝐶𝐶(4,0) a.

Show: ∆𝑆𝑆𝑆𝑆𝑆𝑆 is NOT an equilateral triangle

b.

What type of triangle is ∆𝑆𝑆𝑆𝑆𝑆𝑆?

∆𝑃𝑃𝑃𝑃𝑃𝑃 has vertices at 𝑃𝑃(−2,3), 𝑇𝑇(3,1), and 𝐻𝐻(0, −2).

Determine if ∆𝑃𝑃𝑃𝑃𝑃𝑃 is an equilateral, isosceles, or scalene triangle. Justify your answer with math.

Unit 13: Shapes on a Coordinate Plane

319

3.

Diego is creating a triangular garden and uses the ∆𝑌𝑌𝑌𝑌𝑌𝑌 with vertices at 𝑌𝑌(−8, −2), 𝑉𝑉(−6,3), and 𝑋𝑋(−2, −2) to design the garden. a.

Graph ∆𝑌𝑌𝑌𝑌𝑌𝑌.

b.

Complete the table. If necessary, round to the nearest thousandth.

Line Segment

4.

Draw an isosceles triangle whose base is three more than the height of the triangle and has one vertex at (2, 5).

Side Length

̅​̅​̅​̅ 𝑌𝑌𝑌𝑌

̅​̅​̅​̅ 𝑉𝑉𝑉𝑉 c.

320

̅​̅​̅​̅ 𝑋𝑋𝑋𝑋

Classify the type of triangular garden Diego is creating.

Unit 13: Shapes on a Coordinate Plane

Unit 13, Lesson 7: Centroids of Triangles

13.6.5 Wrap-Up: Reflection Complete each statement.

1.

A barrier to my learning today was …

Benchmark MA.912.GR.3.3 Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles, and quadrilaterals.

Learning Target  2.

I can use coordinate geometry to identify the centroid of a triangle.

I will try to overcome this by …

Unit 13: Shapes on a Coordinate Plane

321

13.7.1 Warm-Up: Human Hamster Wheel A human hamster wheel is shown.

1.

13.7.2 Exploration: Centroids of Triangles 1.

What is the first question that comes to your mind?

∆𝐻𝐻𝐻𝐻𝐻𝐻 has vertices at 𝐻𝐻(−7,3), 𝐾𝐾(2,8), and 𝑅𝑅(5,1).

a.

Complete the table by finding the midpoint of each line segment.

Line Segment

2.

What are two things you notice about the human hamster wheel?

̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅

322

Unit 13: Shapes on a Coordinate Plane

Work

Midpoint

b.

1 2 3 4 5 6 7 c.

Complete the steps to find the centroid of triangle 𝐻𝐻𝐻𝐻𝐻𝐻.

Graph the midpoint of line ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻 and label it 𝑊𝑊.

Using a straightedge, draw the median from 𝑊𝑊 to the opposite vertex, ∠𝐻𝐻𝐻𝐻𝐻𝐻.

Graph the midpoint of line ̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 and label it 𝑆𝑆.

Using a straightedge, draw the median from 𝑆𝑆 to the opposite vertex, ∠𝐾𝐾𝐾𝐾𝐾𝐾.

Graph the midpoint of line ̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 and label it 𝑀𝑀.

Using a straightedge, draw the median from 𝑀𝑀 to the opposite vertex, ∠𝐻𝐻𝐻𝐻𝐻𝐻.

Label the centroid of triangle 𝐻𝐻𝐻𝐻𝐻𝐻 point 𝑃𝑃.

2.

The façade of a house is shown on the coordinate grid.

The façade is represented by ∆𝑊𝑊𝑊𝑊𝑊𝑊 with vertices 𝑊𝑊(−1, 2.5), 𝐻𝐻(4.5, −3), and 𝐾𝐾(−6.5, −3).

What are the coordinates of the centroid of ∆𝐻𝐻𝐻𝐻𝐻𝐻?

Work with your partner to estimate the location of the centroid of the triangular façade.

Unit 13: Shapes on a Coordinate Plane

323

3.

Use the centroids for ∆𝐻𝐻𝐻𝐻𝐻𝐻 and ∆𝑊𝑊𝑊𝑊𝑊𝑊 to complete the table.

Triangle

5.

Average of Average of the the Centroid 𝒙𝒙 −coordinates 𝒚𝒚 −coordinates of the of the Vertices Vertices

Ask two classmates for their conjectures. Record their conjectures and write your summary of their conjectures. You are the only person who should write in the first two columns of the table. Have the person initial next to your summary stating that your summary was correct.

Conjecture

∆𝐻𝐻𝐻𝐻𝐻𝐻

My Summary of Their Conjecture

Initials

∆𝑊𝑊𝑊𝑊𝑊𝑊 4.

Work with your partner to make a conjecture about the relationships of the centroid and the average of the 𝑥𝑥 and 𝑦𝑦 coordinates.

324

6.

Review your conjecture with your partner. If necessary, make any changes.

Unit 13: Shapes on a Coordinate Plane

13.7.3 Collaboration: Centroids of Triangles 1.

∆𝑀𝑀𝑀𝑀𝑀𝑀 has vertices at 𝑀𝑀(0, −3), 𝐵𝐵(8, 0), and 𝐺𝐺(13, −7).

a.

13.7.4 Your Turn!

1.

∆𝑃𝑃𝑃𝑃𝑃𝑃 has vertices at 𝑃𝑃(5,9), 𝐾𝐾(9,3), and 𝑅𝑅(−5,2).

What is the average of the 𝑥𝑥 −coordinates?

b.

What is the average of the 𝑦𝑦 −coordinates?

c.

What are the coordinates of the centroid?

d.

Verify the coordinates of the centroid by drawing the median of each side.

What are the coordinates of the centroid of ∆𝑃𝑃𝑃𝑃𝑃𝑃?

Unit 13: Shapes on a Coordinate Plane

325

2.

∆𝑁𝑁𝑁𝑁𝑁𝑁 has vertices at 𝑁𝑁(−8, −3) and 𝐻𝐻(3, −1) and a centroid at (−2, −4).

What are the coordinates of vertex 𝑇𝑇?

3.

13.7.5 Wrap-Up: Ticket Out the Door 1.

∆𝑇𝑇𝑇𝑇𝑇𝑇 has vertices at 𝑇𝑇(3,6), 𝐶𝐶(10, −2), and 𝐻𝐻(−4,2).

Artemio is finding the centroid of ∆𝑊𝑊𝑊𝑊𝑊𝑊 with vertices at 𝑊𝑊(−8,5), 𝑌𝑌(7,8), and 𝑉𝑉(−2, −3).

His work is shown.

Artemio’s Work 8+7+2 5+8+3 𝑥𝑥 = 𝑦𝑦 = 3 3 𝑥𝑥 =

17 3

17 16

Centroid ( 3 ,

3

)

𝑦𝑦 =

16 3

a.

Write a note to Artemio explaining his mistake.

b.

What are the coordinates of the centroid of ∆𝑊𝑊𝑊𝑊𝑊𝑊?

326

What are the coordinates of the centroid of ∆𝑇𝑇𝑇𝑇𝑇𝑇?

Unit 13: Shapes on a Coordinate Plane

Unit 13, Lesson 8: Classifying Quadrilaterals – Part 1 Benchmark

13.8.1 Warm-Up: Growth Mindset “Examine what is said and not who speaks.”

– African proverb 1.

MA.912.GR.3.2 Given a mathematical context, use coordinate geometry to classify or justify definitions, properties, and theorems involving circles, triangles, or quadrilaterals.

What would I want my teacher to say to me when he/she sees me taking a risk, trying hard, or pushing through mistakes to learn?

Learning Targets 



I can use coordinate geometry to classify quadrilaterals based on their properties in a mathematical or realworld context.

I can draw a quadrilateral on a coordinate plane using given definitions, properties, or theorems.

Unit 13: Shapes on a Coordinate Plane

327

13.8.2 Exploration: Classifying Quadrilaterals Using Properties 1.

3.

Discuss with your partner if knowing a pair of opposite sides are congruent is sufficient to classify the quadrilateral. Summarize your discussion and justify your decision.

4.

Complete the diagram using the different types of quadrilaterals. Each blue box is a subcategory of quadrilaterals. Each green box is a subcategory of the blue box that it is connected to.

Complete the table by filling in the blanks using the word bank. Words may be used more than once, or they may not be used.

Word Bank bisect(s) congruent parallel perpendicular Classification Parallelogram

Opposite sides are _____________ and _________________.

Parallelogram

Diagonals ____________ each other.

Rectangle

Diagonals are __________________.

Rhombus

Diagonals are __________________.

Square Kite Isosceles Trapezoid 2.

Property

Diagonals are ________________ and ___________________.

Exactly one diagonal _____________ the other, and diagonals are _____________________.

Diagonals are ________________ and one pair of sides are ______________.

Discuss with your partner if knowing two consecutive sides are congruent is sufficient to classify a quadrilateral. Summarize your discussion and justify your decision.

328

Quadrilaterals

Isosceles Trapezoid

5.

Which quadrilateral was not included in the diagram?

6.

Draw a box and any necessary arrows to include the missing quadrilateral in the diagram.

Unit 13: Shapes on a Coordinate Plane

13.8.3 Exploration: Classifying Quadrilaterals 1.

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 has vertices at 𝑅𝑅(−6,2), 𝑀𝑀(−4,5), 𝑊𝑊(−1,3), and 𝐵𝐵(−3,0). a.

c.

What do you notice about the side lengths of 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅?

d.

Select all of the classifications that apply to 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅.

Graph 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 on the coordinate plane.

       e.

Isosceles trapezoid Kite Parallelogram Rectangle Rhombus Square Trapezoid

Complete the table.

Diagonal Line Segment

Length

̅​̅​̅​̅​̅ 𝑅𝑅𝑅𝑅 f.

b. Complete the table.

Line Segment

̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀

What do you notice about the diagonal lengths of 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅?

Length

̅​̅​̅​̅​̅ 𝑅𝑅𝑅𝑅

̅​̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 ̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 ̅​̅​̅​̅ 𝐵𝐵𝐵𝐵

Unit 13: Shapes on a Coordinate Plane

329

g.

Complete the statement.

Since 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅

o has o does not have

__________________.

𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 has vertices at 𝑁𝑁(2,3), 𝑇𝑇(3,5), 𝐶𝐶(6,6), and 𝑉𝑉(4,2). a.

Complete the table.

Line Segment ̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

What do you notice about the slopes of 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁?

e.

Select all of the classifications that apply to 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁.

congruent

diagonals, 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 is classified as a(n)

2.

d.

      

Isosceles trapezoid Kite Parallelogram Rectangle Rhombus Square Trapezoid

Length

̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶

̅​̅​̅​̅ 𝑉𝑉𝑉𝑉

b.

What do you notice about the side lengths of 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁?

c.

Complete the table.

Line Segment ̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

Slope

̅​̅​̅​̅ 𝑇𝑇𝑇𝑇 ̅​̅​̅​̅ 𝐶𝐶𝐶𝐶

330

̅​̅​̅​̅ 𝑉𝑉𝑉𝑉

Unit 13: Shapes on a Coordinate Plane

13.8.4 Guided Practice: Classifying Quadrilaterals 1.

𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 has vertices at 𝐺𝐺(−2, −3), 𝑆𝑆(3, −2), 𝑃𝑃(5, −4), and 𝐷𝐷(0, −5).

3.

Use the coordinate grid to complete the following. a. Draw a quadrilateral for which the side lengths are all equal, but the diagonals are not congruent.

Justify that 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 is a parallelogram but not a rectangle.

Show your work.

2.

𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 has vertices at 𝐻𝐻(−2,4), 𝑀𝑀(−2,0), 𝑇𝑇(−6, −1), and 𝐾𝐾(−6,5). Classify 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 and justify your classification using mathematics.

b.

Complete the statement.

This quadrilateral is classified as a _______________.

Unit 13: Shapes on a Coordinate Plane

331

4.

Discuss with your partner how another classmate could use the coordinates of your quadrilateral to verify that your classification is correct. Summarize your discussion.

13.8.5 Your Turn!

1.

𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 has vertices at 𝐾𝐾(1,6), 𝐻𝐻(3,4), 𝐶𝐶(1,1), and 𝑌𝑌(−1,4).

𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 is classified as a _______________ because …

332

Unit 13: Shapes on a Coordinate Plane

2.

Draw a quadrilateral with one set of parallel sides and vertices at point (−4,7) and (−3,5).

13.8.6 Wrap-Up: Error Analysis 1.

𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 has vertices at 𝑌𝑌(−2, 2), 𝑉𝑉(2, 4), 𝑍𝑍(3, 0), and 𝑊𝑊(−1, −2).

Aryana graphed 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 and classified the quadrilateral as a rhombus.

This quadrilateral is classified as a _______________ because …

a.

Do you agree with Aryana?

b.

Justify your answer using properties and showing your work.

Unit 13: Shapes on a Coordinate Plane

333

Unit 13, Lesson 9: Classifying Quadrilaterals – Part 2 Benchmarks MA.912.GR.3.2 Given a mathematical context, use coordinate geometry to classify or justify definitions, properties, and theorems involving circles, triangles, or quadrilaterals.

MA.912.GR.3.3 Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles, and quadrilaterals.

Learning Targets 



Sigsbee Deep Dwight Sigsbee led the first expedition to chart the floor of the Gulf of Mexico. The Sigsbee Deep, the deepest region of the Gulf of Mexico, was named after him. The exact maximum depth is not known, but is estimated to be between 3,750 meters (m) and 4,384 m.

1.

I can use coordinate geometry to classify quadrilaterals based on their properties in a mathematical or realworld context.

I can use coordinate geometry to solve geometric problems involving quadrilaterals in a mathematical or real-world context.

334

13.9.1 Warm-Up: Between Two Numbers

2.

Britton Hill Britton Hill, located in Walton County, is the highest point in the state of Florida.

Britton Hill’s elevation is 105 m.

Panorama Tower Panorama Tower in Miami is the tallest building in Florida. It is 265 m tall.

Complete the inequality using the digits 0 – 9, without repeating, where 𝐵𝐵 = number of Britton Hills and 𝑃𝑃 = number of Panorama Towers.

3750 <

𝐵𝐵 +

𝑃𝑃 < 4384

3750 <

𝐵𝐵 +

𝑃𝑃 < 4384

Use a different set of digits, without repeating.

Unit 13: Shapes on a Coordinate Plane

13.9.2 Collaboration: Real-World Quadrilaterals 1.

Francesco is using computer software to design the layout of a house. The base of the house creates 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 with coordinates 𝐷𝐷(−4, 1), 𝑀𝑀(4, 5), 𝐻𝐻(6, 1), and 𝑇𝑇(−2, −3). a.

b.

Complete the table. Show your work in the table.

Leave answers in exact form.

Diagonal Line Segment

Length

Complete the table. Show your work in the table.

Leave answers in exact form.

Line Segment

̅​̅​̅​̅ 𝐷𝐷𝐷𝐷

Length

̅​̅​̅​̅​̅ 𝐷𝐷𝐷𝐷

̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀

̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀

c.

Finish the statement. 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 is classified as a _____________ because ….

̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

d.

If 1 unit on the coordinate plane equals 5 feet (ft), what are the dimensions of the base of the house?

Round to the nearest thousandth.

̅​̅​̅​̅ 𝑇𝑇𝑇𝑇

Unit 13: Shapes on a Coordinate Plane

335

2.

Jillian is determining which type of tile to place on the floor. She wants a tile in the shape of a parallelogram.

The vertices of the tile options are shown.

Tile A (−11,6.5), (−10,3.5), (−6,6.5), and (−7,9.5) a.

b.

336

Tile B (3,9.5), (7,6), (9,8.5), and (5,11)

Which tile should Jillian pick if she wants a tile in the shape of a parallelogram?

3.

Aharon and Shreya classified 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 with vertices at 𝑅𝑅(−3,7), 𝑁𝑁(−1,7), 𝑃𝑃(−3,2), and 𝐹𝐹(−5,4). Their classifications are shown.

Aharon’s Classification Shreya’s Classification 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 is a square or Quadrilateral 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 has rectangle because the congruent diagonals but diagonals are congruent. cannot be classified as a parallelogram. a.

Whose classification do you agree with? Justify your answer using properties.

b.

Write a note to the person whose classification is incorrect, explaining their mistake.

Explain your choice by listing the properties of parallelograms that were applied.

Unit 13: Shapes on a Coordinate Plane

4.

𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 has vertices at 𝐾𝐾(10, −4), 𝑉𝑉(9, −8), 𝑆𝑆(1, −4), and 𝑊𝑊(6, −2). a.

b.

Show mathematically that 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 is a trapezoid, but not an isosceles trapezoid.

List the properties of an isosceles trapezoid that did not apply to 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾.

5.

Ariyon was graphing square 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀, but mixed up her vertices. She knows that point 𝑀𝑀 has coordinates (3, 5) and point 𝐺𝐺 has coordinates (7, 7). Her two options for points 𝑁𝑁 and 𝑅𝑅 are shown.

Point 𝑵𝑵 (8,3) (9,3)

Point 𝑹𝑹 (5,1) (4,1)

a.

Which coordinates are correct for points 𝑁𝑁 and 𝑅𝑅?

Justify your answer or show your work.

b.

Which properties of a square did you use to determine the correct points 𝑁𝑁 and 𝑅𝑅?

Unit 13: Shapes on a Coordinate Plane

337

13.9.3 Your Turn!

1.

b.

List the properties of a kite that you used.

c.

If 1 unit on the coordinate plane equals 6 inches (in), what are the measurements of the sides of the kite? Round to the nearest thousandth.

𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 has vertices at 𝑆𝑆(−3, −4), 𝐶𝐶(0, −2), 𝑊𝑊(0, −10), and 𝐾𝐾(−3, −8).

𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 is classified as a(n) _______________________ because …

2.

Ziquan is designing a template for his 3D printer. He created two quadrilaterals using his software. 𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻

𝑇𝑇(16, 6), 𝐺𝐺(18, 4), 𝐻𝐻(17, −2), and 𝑁𝑁(14, 4) a.

338

𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴

𝑀𝑀(4, 5), 𝑊𝑊(7, 4), 𝑅𝑅(8, −3), and 𝐵𝐵(3, 2)

Which quadrilateral can be classified as a kite?

Show your work.

Unit 13: Shapes on a Coordinate Plane

Unit 13, Lesson 10: Classifying Quadrilaterals – Part 3

13.9.4 Wrap-Up: Reflection 1.

Place an X on the line in relation to how you feel about today’s learning target. 

I can use coordinate geometry to classify quadrilaterals based on their properties in a mathematical or real-world context.

I need help.

I can’t get started.

I’m getting there.

I understand and have evidence.

Benchmark MA.912.GR.3.2 Given a mathematical context, use coordinate geometry to classify or justify definitions, properties, and theorems involving circles, triangles, or quadrilaterals.

Learning Target 

I can use coordinate geometry to classify quadrilaterals based on their properties in a mathematical or realworld context.

Unit 13: Shapes on a Coordinate Plane

339

13.10.1 Warm-Up: Would You Rather?

13.10.2 Activity: Stations 1.

Would you rather …

Give up watching movies?

OR

Give up watching TV shows?

Complete the table for each question card.

Question

Work

Answer Properties

2.

Explain your choice.

Slopes

1

Side Lengths

Quadrilateral

Diagonal Lengths

340

Unit 13: Shapes on a Coordinate Plane

Question

Work

Answer

Question

Work

Verification

Properties

Slopes

2

Answer

Length ̅​̅​̅​̅​̅​̅ of 𝑾𝑾𝑾𝑾 and ̅​̅​̅​̅ 𝑯𝑯𝑯𝑯

Side Lengths

3

Slopes of ̅​̅​̅​̅​̅​̅ and 𝑾𝑾𝑾𝑾 ̅​̅​̅​̅ 𝑯𝑯𝑯𝑯

Quadrilateral

Diagonal Lengths

Slope of ̅​̅​̅​̅ 𝑪𝑪𝑪𝑪 Using Each Point 𝑺𝑺

Unit 13: Shapes on a Coordinate Plane

341

Question

Work

Answer

Question

Work

Properties

Properties

Slopes

Slopes

4

5

Side Lengths

Side Lengths

Who is correct?

Diagonal Lengths

342

Answer

Quadrilateral

Diagonal Lengths

Unit 13: Shapes on a Coordinate Plane

Question

Work

Answer Which quadrilateral is a rectangle?

Question

Slopes

Work

Answer Why is 𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽 not a square?

Slopes

Properties

6

Side Lengths

Diagonal Lengths

7

Side Lengths

Diagonal Lengths

Unit 13: Shapes on a Coordinate Plane

343

Question

Work

Answer

13.10.3 Wrap-Up: Reflection

Verification 1.

Take a SELFIE by checking the box that best matches what you did today.

Showed my work Slope and Length ̅​̅​̅​̅​̅ of 𝑲𝑲𝑲𝑲

Explained my answers Lots of math vocabulary used Found solutions I persevered through the problem

8

Eliminated careless errors

Slope of ̅​̅​̅​̅​̅ 𝑴𝑴𝑴𝑴 Using Each Point 𝑺𝑺

344

Unit 13: Shapes on a Coordinate Plane

Unit 13, Lesson 11: What Is the Figure?

13.11.1 Warm-Up: Bell Work Jonah is designing a stage in the shape of a trapezoid.

Jonah has created two designs.

Benchmarks MA.912.GR.3.2 Given a mathematical context, use coordinate geometry to classify or justify definitions, properties, and theorems involving circles, triangles, or quadrilaterals.

MA.912.GR.3.3 Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles, and quadrilaterals.

1.

𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽 𝑉𝑉(2, −5), 𝑆𝑆(4, −7), 𝑍𝑍(−2, −7), and 𝑌𝑌(−1, −5)

𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝐵𝐵(−4, −2), 𝐺𝐺(−2, −2), 𝐾𝐾(−7, −7), and 𝑅𝑅(−7, −5)

Determine whether Jonah can use either or both designs for the trapezoidal stage.

• •

Show your work in an organized way.

Write your conclusion explaining the properties used to determine your answer.

Learning Targets   

I can use coordinate geometry to classify triangles and quadrilaterals.

I can use coordinate geometry to justify definitions, properties, or theorems of triangles and quadrilaterals.

I can use coordinate geometry to solve mathematical problems involving triangles and quadrilaterals.

Unit 13: Shapes on a Coordinate Plane

345

Match 3

13.11.2 Activity: Card Sort Match each image, classification, coordinates, and property that represent the same polygon. Record your matches in the table by writing the letter or number of each card and showing your justifications.

Match 1

Match 4

Image:

Image:

Coordinates:

Coordinates:

Classification:

Classification:

Property:

Property:

Justification:

Justification:

Match 2

Image:

Image:

Coordinates:

Coordinates:

Classification:

Classification:

Property:

Property:

Justification:

Justification:

Match 5

346

Match 6

Image:

Image:

Coordinates:

Coordinates:

Classification:

Classification:

Property:

Property:

Justification:

Justification:

Unit 13: Shapes on a Coordinate Plane

Match 7

13.11.3 Wrap-Up: Reflection

Match 8

Image:

Image:

Coordinates:

Coordinates:

Classification:

Classification:

Property:

Property:

Justification:

Justification:

1.

Complete the table.

Today I deserve

for …

Today I

Match 9

Match 10

Image:

Image:

Coordinates:

Coordinates:

Classification:

Classification:

Property:

Property:

Justification:

Justification:

by doing …

Today I was

by …

Unit 13: Shapes on a Coordinate Plane

347

Unit 13, Lesson 12: Congruence and Similarity

13.12.1 Warm-Up: Notice and Wonder Brickley, the LEGO sea serpent in Lake Buena Vista at Disney Springs near Orlando, Florida, was made in 1997 from 170,000 LEGO bricks and is 30 feet long.

Benchmark

MA.912.GR.3.3 Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles, and quadrilaterals.

Learning Target 

I can use coordinate geometry to justify congruence and similarity in triangles and quadrilaterals.

348

1.

What do you notice?

2.

What do you wonder?

Unit 13: Shapes on a Coordinate Plane

13.12.2 Exploration: Congruence and Similarity 1.

2.

Complete the table.

Parallelogram 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 has vertices at 𝐷𝐷(−2,4), 𝐾𝐾(−4,1), 𝑇𝑇(−9,3), and 𝑆𝑆(−7,6).

Line Segment

Complete the table.

Line Segment

Length

̅​̅​̅​̅ 𝑅𝑅𝑅𝑅

̅​̅​̅​̅ 𝐶𝐶𝐶𝐶

̅​̅​̅​̅ 𝐾𝐾𝐾𝐾

̅​̅​̅​̅ 𝑆𝑆𝑆𝑆

Length

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊

̅​̅​̅​̅ 𝐷𝐷𝐷𝐷

̅​̅​̅​̅ 𝑇𝑇𝑇𝑇

Parallelogram 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 has vertices at 𝑊𝑊(0,1), 𝑅𝑅(2, −2), 𝐶𝐶(7,0), and 𝑁𝑁(5,3).

̅​̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

3.

What do you notice about the lengths of the line segments in parallelograms 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 and 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊?

4.

Complete the statement.

Parallelogram 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 may be

o congruent o similar

to parallelogram 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 because all corresponding sides are

o proportional. o congruent.

Unit 13: Shapes on a Coordinate Plane

349

5.

Work with your partner to describe how the parallelograms are congruent or similar using transformations. State which image you used as the pre-image.

7.

Isosceles trapezoid 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 has vertices at 𝑊𝑊(4,12), 𝑉𝑉(6,6), 𝑌𝑌(10,4), and 𝑍𝑍(16,6).

Complete the table.

Line Segment

Length

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 ̅​̅​̅​̅ 𝑉𝑉𝑉𝑉 ̅​̅​̅​̅ 𝑌𝑌𝑌𝑌

6.

Isosceles trapezoid 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 has vertices at 𝑀𝑀(2, −6), 𝑃𝑃(3, −3), 𝐻𝐻(5, −2), and 𝑁𝑁(8, −3).

8.

What do you notice about the lengths of the line segments in isosceles trapezoids 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 and 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊?

9.

Complete the statement.

Complete the table.

Line Segment

̅​̅​̅​̅​̅ 𝑍𝑍𝑍𝑍

Length

̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 ̅​̅​̅​̅ 𝑃𝑃𝑃𝑃

̅​̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

̅​̅​̅​̅​̅ 𝑁𝑁𝑁𝑁

Isosceles trapezoid 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 is

to

isosceles trapezoid 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 because all corresponding sides are

350

o congruent o similar

o proportional. o congruent.

Unit 13: Shapes on a Coordinate Plane

10. Work with your partner to describe how the trapezoids are congruent or similar using transformations. State which image you used as the pre-image.

13.12.3 Guided Practice: Congruence and Similarity 1.

Right triangle 𝐶𝐶𝐶𝐶𝐶𝐶 has vertices at 𝐶𝐶(−2,5), 𝐾𝐾(−8,5), and 𝑅𝑅(−8,2). Right triangle 𝐷𝐷𝐷𝐷𝐷𝐷 has vertices at 𝐷𝐷(−2, −4), 𝑃𝑃(4, −4), and 𝑀𝑀(4, −1). a.

Determine if ∆𝐶𝐶𝐶𝐶𝐶𝐶 and ∆𝐷𝐷𝐷𝐷𝐷𝐷 are congruent, similar, or neither.

b.

What definition, theorem, or property justifies your conclusions?

c.

Describe the transformations that would show that the figures are congruent or that they are similar.

State which image you used as the pre-image.

Unit 13: Shapes on a Coordinate Plane

351

d.

Ask a classmate for their description. Record their description and write your summary of their description. You are the only person who should write in the first two columns of the table. Have the person initial next to your summary stating that your summary was correct.

Description

My Summary of Their Description

2.

Parallelogram 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 has vertices at 𝐻𝐻(5, −1), 𝐺𝐺(2,3), 𝑁𝑁(9,3), and 𝐶𝐶(12, −1). Parallelogram 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 has vertices at 𝑉𝑉(−1,1.5), 𝑌𝑌(0.5,4), 𝑍𝑍(−3,4), and 𝑊𝑊(−4.5,1.5). a.

Initials

Shin and Kendal are determining if parallelograms 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 and 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 are congruent, similar, or neither.

Their work is shown.

Shin’s Work

Kendal’s Work

Parallelograms 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 and 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 are similar because 𝑍𝑍𝑍𝑍 𝑉𝑉𝑉𝑉 = 𝐻𝐻𝐻𝐻 𝑁𝑁𝑁𝑁

Parallelograms 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 and 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 are neither because 𝑍𝑍𝑍𝑍 𝑌𝑌𝑌𝑌 ≠ 𝐺𝐺𝐺𝐺 𝑁𝑁𝑁𝑁

b.

Whose work do you agree with?

c.

Write a note to the person who is incorrect explaining their mistake.

Means I Start To Acquire Knowledge Experience Skills

352

Unit 13: Shapes on a Coordinate Plane

3.

13.12.4 Your Turn!

Complete the table by determining if the given polygons are congruent, similar, or neither.

1.

Kite 𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋: 𝑋𝑋(−7,6), 𝑍𝑍(−6,9), 𝑊𝑊(−3,10), and 𝑌𝑌(−1,4) Kite 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹: 𝐹𝐹(2,3), 𝐺𝐺(1,6), 𝐷𝐷(−2,7), and 𝐾𝐾(−4,1) o Congruent o Similar o Neither

∆𝑆𝑆𝑆𝑆𝑆𝑆: 𝑆𝑆(−6,8), 𝑇𝑇(−10,5), and 𝐾𝐾(−6,1) ∆𝑁𝑁𝑁𝑁𝑁𝑁: 𝑁𝑁(3,7.5), 𝑅𝑅(−3,3), and 𝐶𝐶(3, −3) o o o

2.

Congruent Similar Neither

Trapezoid 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀: 𝑀𝑀(3,4), 𝑃𝑃(7,6), 𝑊𝑊(10,5), and 𝑌𝑌(4,2) Trapezoid 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿: 𝐿𝐿(−2,6), 𝐾𝐾(−8,9), 𝐺𝐺(−14,7), and 𝐻𝐻(−4,2) o Congruent o Similar o Neither

Unit 13: Shapes on a Coordinate Plane

353

Unit 13, Lesson 13: Finding Perimeter and Area on the Coordinate Plane

13.12.5 Wrap-Up: Cool Down 1.

Your friend is absent today and will need the notes from class. Write a quick note to teach them how to use coordinate geometry to justify congruence and similarity in polygons.

Benchmark MA.912.GR.3.4 Use coordinate geometry to solve mathematical and real-world problems on the coordinate plane involving the perimeter or area of polygons.

Learning Target 

354

I can find the perimeter and area of polygons and composite figures drawn on the coordinate plane.

Unit 13: Shapes on a Coordinate Plane

13.13.1 Warm-Up: The Dancing House The Dancing House is located in Prague, Czech Republic, and was designed in 1992. The style of this building is known as deconstructivist architecture due to its unusual shape.

13.13.2 Exploration: Perimeter and Area 1.

Trapezoid 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 has vertices at 𝐵𝐵(1,9), 𝐺𝐺(5,9), 𝑀𝑀(2,5), and 𝐻𝐻(−4,5).

a.

1.

2.

What is the first question that comes to your mind?

Write a question that a 5th grader could answer using the photo.

Graph 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵.

b. Complete the table.

Line Segment

Length

̅​̅​̅​̅ 𝐵𝐵𝐵𝐵

̅​̅​̅​̅​̅ 𝐺𝐺𝐺𝐺

̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀 ̅​̅​̅​̅ 𝐻𝐻𝐻𝐻

Unit 13: Shapes on a Coordinate Plane

355

c.

What is the perimeter of 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵?

d.

Draw a diagonal line from 𝐵𝐵 to 𝑀𝑀 to make ∆𝐵𝐵𝐵𝐵𝐵𝐵 and ∆𝐵𝐵𝐵𝐵𝐵𝐵.

e.

Parallelogram 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 has vertices at 𝐺𝐺(5,9), 𝑃𝑃(5,6), 𝐾𝐾(2,2), and 𝑀𝑀(2,5).

Find the area of ∆𝐵𝐵𝐵𝐵𝐵𝐵 and ∆𝐵𝐵𝐵𝐵𝐵𝐵.

∆𝐵𝐵𝐵𝐵𝐵𝐵 =__________

f.

2.

∆𝐵𝐵𝐵𝐵𝐵𝐵 =__________

What is the area of 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵?

a.

Graph 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺.

b. Complete the table.

Line Segment ̅​̅​̅​̅ 𝐺𝐺𝐺𝐺

̅​̅​̅​̅ 𝑃𝑃𝑃𝑃

̅​̅​̅​̅​̅ 𝐾𝐾𝐾𝐾 ̅​̅​̅​̅​̅ 𝑀𝑀𝑀𝑀

356

Unit 13: Shapes on a Coordinate Plane

Length

c.

d.

3.

What is the perimeter of 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺?

A composite figure is a two- or threedimensional figure that can be decomposed into smaller figures.

What is the area of 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺?

The figure shown is created by trapezoid 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 and parallelogram 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺.

a.

b.

Explain why this figure is considered a composite figure.

c.

How could you find the perimeter of a composite figure?

d.

What is the perimeter of composite figure 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵?

e.

Explain why the perimeter of the composite figure is not the sum of the perimeters of the parallelogram and the trapezoid.

f.

What is the area of 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵?

What do you notice about this figure?

Unit 13: Shapes on a Coordinate Plane

357

13.13.3 Try It: Perimeter and Area 1.

b.

What is the perimeter of 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌?

c.

Draw line 𝑋𝑋𝑋𝑋 to create ∆𝑌𝑌𝑌𝑌𝑌𝑌 and ∆𝑉𝑉𝑉𝑉𝑉𝑉.

Kite 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 is shown with vertices at 𝑌𝑌(3,6), 𝑉𝑉(5,3), 𝑊𝑊(3, −2), and 𝑋𝑋(1,3).

d.

Find the area of each triangle.

∆𝑌𝑌𝑌𝑌𝑌𝑌 =__________

e. a. Complete the table.

Line Segment

What is the area of 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌?

Length

̅​̅​̅​̅ 𝑌𝑌𝑌𝑌

̅​̅​̅​̅​̅ 𝑉𝑉𝑉𝑉

̅​̅​̅​̅​̅ 𝑊𝑊𝑊𝑊 ̅​̅​̅​̅ 𝑋𝑋𝑋𝑋

358

∆𝑉𝑉𝑉𝑉𝑉𝑉 =__________

Unit 13: Shapes on a Coordinate Plane

2.

The composite figure shown is created by ∆𝑁𝑁𝑁𝑁𝑁𝑁 and parallelogram 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅. 𝑁𝑁(3,6), 𝐶𝐶(11,6), 𝐺𝐺(11,3), 𝑆𝑆(6,0), and 𝑅𝑅(6,3).

a.

13.13.4 Wrap-Up: Cool Down 1.

What is something that is now “squared away” in your mind about finding the perimeter and area of polygons and composite figures drawn on a coordinate plane?

2.

What are three “points” that are important about finding the perimeter and area of polygons and composite figures drawn on a coordinate plane?

3.

What is something that is still “circling” your brain about finding the perimeter and area of polygons and composite figures drawn on a coordinate plane?

Complete the table. Round your answer to the nearest thousandth.

Figure

Perimeter

Area

∆𝑁𝑁𝑁𝑁𝑁𝑁 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅

b.

What is the perimeter of the composite figure?

c.

What is the area of the composite figure?

Unit 13: Shapes on a Coordinate Plane

359

Unit 13, Lesson 14: Using Coordinate Geometry to Solve Real-World Problems

13.14.1 Warm-Up: Bell Work 1.

The composite figure shown is created by ∆𝑀𝑀𝑀𝑀𝑀𝑀 and trapezoid 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷. 𝑀𝑀(4, −2), 𝐾𝐾(8.5, −2), 𝑅𝑅(4, −7), 𝐻𝐻(1, −9), and 𝐷𝐷(1, −1).

Benchmarks MA.912.GR.3.2 Given a mathematical context, use coordinate geometry to classify or justify definitions, properties, and theorems involving circles, triangles, or quadrilaterals.

MA.912.GR.3.3 Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles, and quadrilaterals.

MA.912.GR.3.4 Use coordinate geometry to solve mathematical and real-world problems on the coordinate plane involving perimeter or area of polygons.





I can use coordinate geometry to classify triangles and quadrilaterals.

I can use coordinate geometry to solve real-world problems on the coordinate plane involving perimeter or area of polygons.

I can use coordinate geometry to solve real-world geometric problems involving circles, triangles, and quadrilaterals.

360

Complete the table. If necessary, round to the nearest thousandth.

Figure Perimeter Area ∆𝑀𝑀𝑀𝑀𝑀𝑀

Learning Targets 

a.

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷

b.

What is the perimeter of the composite figure? If necessary, round to the nearest thousandth.

c.

What is the area of the composite figure? If necessary, round to the nearest thousandth.

Unit 13: Shapes on a Coordinate Plane

13.14.2 Collaboration: Real-World Problems 1.

Lana is designing the front of a doghouse. The computer software generated coordinate points where each unit equals a foot. The roof has vertices at 𝑀𝑀(7,9), 𝑃𝑃(11,5), and 𝑅𝑅(3,5). The front wall has vertices at 𝑃𝑃(11,5), 𝐾𝐾(11,1), 𝐷𝐷(3,1), and 𝑅𝑅(3,5). The door entrance has vertices at 𝑍𝑍(9,1), 𝑌𝑌(5,1), 𝑊𝑊(6,4), and 𝑉𝑉(8,4).

a.

What type of triangle is the roof, ∆𝑀𝑀𝑀𝑀𝑀𝑀? Justify your answer.

b.

What is the area of ∆𝑀𝑀𝑀𝑀𝑀𝑀?

c.

Clay tiles cost $8 per square foot. How much will it cost Lana to put clay tiles on the given face of the doghouse’s roof?

d.

What type of quadrilateral is the door, 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊?

Justify your answer.

e.

Lana is placing a frame around the door, 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊.

How many feet of wood will she need? If necessary, round to the nearest thousandth.

f.

What type of quadrilateral is the front wall, 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅?

Justify your answer.

Unit 13: Shapes on a Coordinate Plane

361

g.

Lana is painting the front wall, 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅, excluding the door, 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊. What is the area of the front wall Lana is painting?

2.

Justin is creating a layout for his garden in the shape of a quadrilateral, 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺, with vertices 𝐺𝐺(1,8), 𝑊𝑊(11,8), 𝐷𝐷(11,0), and 𝑃𝑃(1,0). a.

Classify 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺. Justify your answer.

Justin created three sections in the garden: 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇, 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿, and ∆𝐾𝐾𝐾𝐾𝐾𝐾. h.

362

b.

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 has vertices at 𝑇𝑇(2,7), 𝐻𝐻(10,7), 𝐿𝐿(8,3), and 𝑀𝑀(3,3). Classify 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇. Justify your answer.

c.

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 has vertices at 𝐿𝐿(8,3), 𝐾𝐾(7,1), 𝑆𝑆(2,1), and 𝑀𝑀(3,3). Classify 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿. Justify your answer.

d.

∆𝐾𝐾𝐾𝐾𝐾𝐾 has vertices at 𝐾𝐾(7,1), 𝑁𝑁(9,1), and 𝑅𝑅(9,5).

Classify ∆𝐾𝐾𝐾𝐾𝐾𝐾. Justify your answer.

Paint costs $0.55 per square foot. Excluding the door, how much will it cost to paint the front wall?

Unit 13: Shapes on a Coordinate Plane

The layout of the garden is shown.

f.

Complete the table. If necessary, round to the nearest thousandth.

Section

Perimeter

Area

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 ∆𝐾𝐾𝐾𝐾𝐾𝐾

e.

If one square on the grid represents one square foot, how many feet of fencing will Justin need to go around 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺?

g.

What is the perimeter of the composite figure made by the three sections of the garden? If necessary, round to the nearest thousandth.

h.

What is the area of the three sections of the garden? If necessary, round to the nearest thousandth.

i.

If a 40 pound (lb) bag of soil covers 12 square feet, how many bags of soil will Justin need to cover the three sections of the garden?

Unit 13: Shapes on a Coordinate Plane

363

3.

Edwin started creating a new logo design for the company office using the images shown.

a.

Triangle

To determine how to place the logo on the wall in the company’s office, he uploaded his design to a computer program and got the following coordinates. 𝑀𝑀(−3,2)

5 𝐺𝐺(− , 4) 3

𝑅𝑅(−9,6) 31 𝐻𝐻(− , 4) 3

𝑃𝑃(−3.8,4)

𝐾𝐾(−5.3,7.5)

𝑇𝑇(3,11)

𝐴𝐴(−6,4)

Circle 𝐴𝐴 has points 𝑅𝑅, 𝑀𝑀, and 𝐾𝐾 on the circle. Lines 𝑇𝑇𝑇𝑇 and 𝐻𝐻𝐻𝐻 are tangent to circle 𝐴𝐴. Lines 𝑅𝑅𝑅𝑅 and 𝐻𝐻𝐻𝐻 are secant to circle 𝐴𝐴. Line segment 𝐾𝐾𝐾𝐾 is perpendicular to line 𝑇𝑇𝑇𝑇.

364

Complete the table by finding the area of each triangle.

Work

Area

∆𝐴𝐴𝐴𝐴𝐴𝐴 ∆𝐴𝐴𝐴𝐴𝐴𝐴 ∆𝑃𝑃𝑃𝑃𝑃𝑃 ∆𝑇𝑇𝑇𝑇𝑇𝑇 b.

Explain how to use ∆𝑃𝑃𝑃𝑃𝑃𝑃 and ∆𝑇𝑇𝑇𝑇𝑇𝑇 to find the area of 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇.

c.

What is the area of 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇? If necessary, round to the nearest thousandth.

d.

What is the total area of Edwin’s design? If necessary, round to the nearest thousandth.

Unit 13: Shapes on a Coordinate Plane

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

13.14.3 Wrap-Up: Reflection Circle the icon that best describes your understanding of each.

I need help.

I got it.

I could help someone.

Using coordinate geometry to determine if lines are parallel or perpendicular.

Using coordinate geometry to partition a line segment or find the weighted average.

Using coordinate geometry to classify triangles or quadrilaterals.

Using coordinate geometry to justify definitions, properties, and theorems involving circles, triangles or quadrilaterals.

Using coordinate geometry to solve mathematical and realworld geometric problems involving lines, circles, triangles, and quadrilaterals.

Using coordinate geometry to solve mathematical and realworld involving perimeter or area of polygons.

Unit 13: Shapes on a Coordinate Plane

365

Benchmark Index MA.912.GR.1.1

Lesson Lesson Lesson Lesson

2.1, 3.1, 3.4, 3.7,

Lesson Lesson Lesson Lesson

2.7, 3.2, 3.5, 3.8,

Lesson Lesson Lesson Lesson

2.8, 3.3, 3.6, 3.9

MA.912.GR.1.2

Lesson Lesson Lesson Lesson

5.1, Lesson 5.2, Lesson 5.3, 5.5, Lesson 5.6, Lesson 5.7, 5.8, Lesson 6.7, Lesson 6.8, 6.9

MA.912.GR.1.3

Lesson 11.5, Lesson 11.6, Lesson 11.7, Lesson 11.8, Lesson 11.9

MA.912.GR.1.4

Lesson 8.1, Lesson 8.2, Lesson 8.3, Lesson 8.4, Lesson 8.6

MA.912.GR.1.5

Lesson 8.5, Lesson 8.6, Lesson 8.7

MA.912.GR.1.6

Lesson 5.4, Lesson 5.7, Lesson 5.8, Lesson 5.9, Lesson 5.10, Lesson 6.10, Lesson 6.11, Lesson 6.12, Lesson 6.13

MA.912.GR.2.1

Lesson 4.1, Lesson 4.2, Lesson 4.4, Lesson 4.5, Lesson 5.1, Lesson 6.1, Lesson 6.2

MA.912.GR.2.2

Lesson 1.1, Lesson 1.2, Lesson 1.3, Lesson 1.4, Lesson 4.3, Lesson 6.3, Lesson 6.4

MA.912.GR.2.3

Lesson 1.5, Lesson 1.6, Lesson 4.3, Lesson 4.6, Lesson 4.7, Lesson 4.8, Lesson 5.4, Lesson 6.2, Lesson 6.3, Lesson 6.4, Lesson 6.6

MA.912.GR.2.4

Lesson 1.H1

MA.912.GR.2.5

MA.912.GR.4.1

Lesson 9.3

MA.912.GR.4.2

Lesson 9.4

MA.912.GR.4.3

Lesson 9.10, Lesson 9.11

MA.912.GR.4.4

Lesson 9.1, Lesson 9.2

MA.912.GR.4.5

Lesson 9.5, Lesson 9.6, Lesson 9.7

MA.912.GR.4.6

Lesson 9.8, Lesson 9.9

MA.912.GR.5.1

Lesson 1.8, Lesson 1.9, Lesson 5.1, Lesson 5.2, Lesson 5.3

MA.912.GR.5.2

Lesson 1.10, Lesson 1.11, Lesson 1.12, Lesson 11.7

MA.912.GR.5.3

Lesson 11.1, Lesson 11.2

MA.912.GR.5.4

Lesson 11.H1

MA.912.GR.5.5

Lesson 10.H1

MA.912.GR.6.1

Lesson 12.1, Lesson 12.2, Lesson 12.3, Lesson 12.4, Lesson 12.5, Lesson 12.6

MA.912.GR.6.2

Lesson 10.5, Lesson 10.6, Lesson 10.7, Lesson 10.8, Lesson 10.9

MA.912.GR.6.3

Lesson 11.3, Lesson 11.4

MA.912.GR.6.4

Lesson 10.1, Lesson 10.2, Lesson 10.3, Lesson 10.4

Lesson 4.4, Lesson 4.5, Lesson 6.3, Lesson 6.5

MA.912.GR.6.5

Lesson 6.H1

MA.912.GR.2.6

Lesson 1.7, Lesson 4.6, Lesson 4.7, Lesson 4.8, Lesson 5.4

MA.912.GR.7.2

Lesson 12.7, Lesson 12.8, Lesson 13.3

MA.912.GR.2.7

Lesson 5.H1

MA.912.GR.7.3

Lesson 12.8, Lesson 12.9

MA.912.GR.2.8

Lesson 1.7, Lesson 6.6

MA.912.LT.4.3

Lesson 2.4, Lesson 2.5, Lesson 2.6, Lesson 3.5

MA.912.GR.2.9

Lesson 6.H3

MA.912.LT.4.8

Lesson 6.H2

MA.912.GR.3.1

Lesson 13.5 Lesson 13.1, Lesson 13.2, Lesson 13.3, Lesson 13.6, Lesson 13.8, Lesson 13.9, Lesson 13.10, Lesson 13.11, Lesson 13.14

MA.912.LT.4.10

MA.912.GR.3.2

Lesson Lesson Lesson Lesson Lesson

MA.912.GR.3.3

Lesson 13.2, Lesson 13.3, Lesson 13.4, Lesson 13.7, Lesson 13.9, Lesson 13.11, Lesson 13.12, Lesson 13.14

MA.912.T.1.1

Lesson 7.1, Lesson 7.2, Lesson 7.3, Lesson 7.4, Lesson 7.7

MA.912.T.1.2

Lesson 7.1, Lesson 7.2, Lesson 7.3, Lesson 7.4, Lesson 7.5, Lesson 7.6, Lesson 7.7, Lesson 7.8, Lesson 7.9

MA.912.GR.3.4

366

Lesson 13.2, Lesson 13.13, Lesson 13.14

2.1, Lesson 2.2, Lesson 2.8, Lesson 3.1, Lesson 5.5, Lesson 5.6, Lesson 8.1, Lesson 8.2, Lesson 11.6, Lesson 11.8

2.3, 3.8, 5.7, 8.3,

Lesson Lesson Lesson Lesson

2.7, 3.9, 5.8, 8.5,

Benchmark Index MA.912.T.1.3

Lesson 7.H1, Lesson 7.H2

MA.912.T.1.4

Lesson 7.H3

MA.K12.MTR.1.1

embedded throughout

MA.K12.MTR.2.1

embedded throughout

MA.K12.MTR.3.1

embedded throughout

MA.K12.MTR.4.1

embedded throughout

MA.K12.MTR.5.1

embedded throughout

MA.K12.MTR.6.1

embedded throughout

MA.K12.MTR.7.1

embedded throughout

ELA.K12.EE.1.1

embedded throughout

ELA.K12.EE.2.1

embedded throughout

ELA.K12.EE.3.1

embedded throughout

ELA.K12.EE.4.1

embedded throughout

ELA.K12.EE.5.1

embedded throughout

ELA.K12.EE.6.1

embedded throughout

ELD.K12.ELL.MA.1 embedded throughout

367

Appendix C: 6-12 Mathematics Glossary absolute value

bar graph

A visual display of categorical data values where each category is represented by a bar whose height represents the number in that category. Bar graphs can be represented vertically or horizontally.

base (of an exponent)

The number used as a factor in exponential form.

bivariate data

Data that measures two characteristics of a population.

box plot

A plot displaying the spread or distribution of a data set using a five number summary, the minimum, lower quartile, median, upper quartile and maximum. It is also called a boxand-whisker plot.

capacity

Angles are formed wherever two lines, segments or rays intersect. Angles are measured in degrees.

The amount of space that can be filled in a container. Both capacity and volume are used to measure three-dimensional spaces; however, capacity usually refers to fluid measures, whereas volume is measured in cubic units.

categorical data

The measure, in square units, of the inside region of a closed two-dimensional figure.

A type of data which is divided into groups.

Categorical data are qualitative.

central angle

An angle that has its vertex at the center of a circle with radii as its sides.

circle graph

A visual display of categorical data. The whole set of data is represented by the circle and its interior. The categories are represented by fractional parts of the circle. Also called a pie chart.

circumference

The distance around a circle.

circumscribed circle

The smallest circle that includes a plane figure.

If the figure is a polygon, then the circle must contain all of the vertices of the polygon. Not every polygon has a circumscribed circle, but all triangles and all regular polygons have circumscribed circles.

additive identity property

Refer to Properties of Operations, Equality and Inequality (Appendix D).

additive inverse property

Refer to Properties of Operations, Equality and Inequality (Appendix D).

addition property of equality

Refer to Properties of Operations, Equality and Inequality (Appendix D).

addition property of inequality

Refer to Properties of Operations, Equality and Inequality (Appendix D).

algorithm

A step-by-step way to solve a problem.

analytic geometry

The branch of mathematics that uses functions and relations to study geometric phenomena.

angle (∠) area arithmetic sequence

association

368

A number’s distance from zero (0) on a number line. Distance is expressed as a positive value.

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

A sequence of numbers in which each consecutive pair of numbers has a common difference.

A way to describe the form, direction or strength of the relationship between the two variables in a bivariate data set. For numerical data, descriptions include linear or nonlinear; positive or negative; strong or weak. For categorical data, descriptions include strong or weak.

associative property

Refer to Properties of Operations, Equality and Inequality (Appendix D).

axes (of a graph)

The horizontal and vertical number lines used in a coordinate plane system.

Appendix C: 6-12 Mathematics Glossary cluster (data)

Data that are in a close group on a scatter plot or univariate numerical data that have similar values.

coefficient

The number or constant that multiplies a variable in an algebraic expression. If no number is specified, the coefficient is 1.

commutative property (of addition or multiplication)

Refer to Properties of Operations, Equality and Inequality (Appendix D).

complementary angles

Two angles whose measures sum to 90°.

composite number

A whole number greater than 1 that has at least one whole-number factor other than one and itself.

compound inequality

A conjunction of two or more inequalities.

concave polygon

A polygon with one or more diagonals that have points outside the polygon. See convex polygon.

conditional relative frequency

The ratio of a joint relative frequency and a marginal relative frequency. Equivalently, the ratio of a relative frequency and a marginal frequency.

cone

A three-dimensional figure with a circular base and an apex that is connected to the base by a collection of line segments that form a curved surface.

congruent

Having exactly the same shape and size.

Equivalently, two figures are congruent if one can be mapped to the other using a rigid transformation.

constant of proportionality

The constant value of the ratio of two proportional quantities.

converse of Pythagorean Theorem

If the lengths 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 of the three sides of a triangle satisfy the relationship 𝑎𝑎! + 𝑏𝑏! = 𝑐𝑐 ! , then the triangle is a right triangle.

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS convex polygon

A polygon with each interior angle measuring less than 180°. All diagonals of a convex polygon lie inside the polygon. See concave polygon.

coordinate

A number used to locate a point on a number line. One of the numbers in an ordered pair, or triple, that locates a point on a coordinate plane or in coordinate space, respectively.

coordinate plane

A plane determined by two perpendicular number lines called axes. The axes intersect at the origin. Each point in the coordinate plane is represented by a pair of coordinates that represent the direction and distance from each axis. The origin is represented by the coordinate pair (0,0).

corresponding angles

Angles that are in the same position on two parallel lines in relation to a transversal.

cube

A rectangular prism with six congruent square faces.

customary units

The units of measure used in the United States.

• Customary units for length include inches, feet, yards, and miles.

• Customary units for weight include ounces, pounds, and tons.

• Customary units for volume include cubic inches, cubic feet, and cubic yards.

• Customary units for capacity include fluid ounces, cups, pints, quarts, and gallons.

A figure containing two congruent, parallel, circular bases whose edges are connected by a cylinder (circular) curved surface. The net of the cylinder consists of a parallelogram and two circles. data

Values that are collected together for reference or analysis.

diameter

A line segment from any point on the circle passing through the center to another point on the circle.

369

Appendix C: 6-12 Mathematics Glossary dilation

A proportional increase or decrease in size in all directions.

distributive property

Refer to Properties of Operations, Equality and Inequality (Appendix D).

domain

The complete set of possible values of the input of a function or relation. The domain may vary depending on the context. See range (of a relation or function).

equilateral triangle

A triangle with three equal-length sides and three 60° interior angles. Also known as an equiangular triangle.

estimation

The use of methods to determine a reasonably accurate approximation, without calculating an exact answer.

geometric sequence

A sequence of numbers in which each consecutive pair of numbers has a common ratio.

greatest common factor (GCF) of two or more whole numbers

The largest whole number that evenly divides the given whole numbers.

histogram

A visual display of numerical data using bars along a number line with no spaces between the bars. The height of each bar represents either the frequency or relative frequency of data within that interval.

hypotenuse

The longest side of a right triangle; the side opposite the right angle.

identity property of addition

Refer to Properties of Operations, Equality and Inequality (Appendix D).

event

A set of possible outcomes resulting from an experiment. In general, an event is any subset of a sample space.

exponent (exponential form)

The number of times the base occurs as a factor.

identity property of multiplication

Refer to Properties of Operations, Equality and Inequality (Appendix D).

exponential function

An exponential function is a function with a constant percent rate of change.

inscribed angle in a circle

An angle which is formed in the interior of a circle when two chords share an endpoint.

experimental probability

The ratio of the number of times an event occurs to the total number of trials or times the activity is performed. Also called empirical probability.

inscribed circle

The largest possible circle that can be contained in a plane figure. If the plane figure is a polygon, then the circle must be tangent to all of the sides of the polygon. Not every polygon has an inscribed circle, but all triangles and all regular polygons have inscribed circles.

inscribed polygon in a circle

A polygon which has all of its vertices on a circle.

integers

Whole numbers and their opposites.

frequency table

function

function notation

370

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

A table that shows how often each item, number, or range of numbers occurs in a set of data.

A mathematical relation for which each element of the domain corresponds to exactly one element of the range.

A notation that describes a function. For a function 𝑓𝑓 when 𝑥𝑥 is a member of the domain, the symbol 𝑓𝑓(𝑥𝑥) denotes the corresponding member of the range.

Appendix C: 6-12 Mathematics Glossary intercept

The value of a variable when all other variables in the equation equal 0. On a graph, the values where a function crosses an axis.

interquartile range (IQR)

A measure of variation in a set of numerical data, the interquartile range is the distance between the first and third quartiles of the data set. See quartile and box plot.

inverse functions irrational number joint frequency

Two functions, 𝑦𝑦 = ℎ(𝑥𝑥) and 𝑥𝑥 = 𝑔𝑔(𝑦𝑦), are said to be inverses when 𝑔𝑔(ℎ(𝑥𝑥)) = 𝑥𝑥 and ℎ(𝑔𝑔(𝑦𝑦)) = 𝑦𝑦. The function inverse to 𝑓𝑓(𝑥𝑥) is denoted 𝑓𝑓 "# (𝑥𝑥).

A real number that cannot be expressed as a ratio of two integers.

In a two-way table, joint frequency is the number of times a combination of two conditions occurs.

joint relative frequency

Joint relative frequency is the ratio of the joint frequency and the total number of data points.

least common multiple (LCM)

The lowest number that is a multiple of two or more given numbers.

line of fit

A line drawn on a scatter plot to estimate the relationship between two sets of data. Also known as a trend line. See scatter plot.

line graph

A graph that displays numerical data using connected line segments.

line plot

A visual display of data values where each data value is shown as a dot or mark above a number line. Also known as a dot plot.

linear expression (or linear equation)

A polynomial expression or equation that contains a term of degree 1, but no term of higher degree.

linear function

A function that has a constant rate of change.

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

line of symmetry

A line that divides a figure into two congruent parts, so that the reflection of either part across the line maps precisely onto the other part.

matrix

A rectangular array of numbers or variables.

mean

The arithmetic average of a set of numbers. It is a measure of central tendency.

measures of center

A numerical value used to describe the overall clustering of data in a set, or the overall central value of a set of data. The three most common measures of central tendency are the mean, median, and mode.

measures of variability

A numerical value that measures how much a data set varies from a central value.

median

The middle of an ordered list of values. If the list has an odd number of values, it is the middle value of that list. If the list has an even number of values, it is the average of the two middle values. It is a measure of central tendency.

metric units

The units of measure used in most of the world. Like the decimal system, the metric system uses the base 10.

• Metric units for length include millimeters, centimeters, meters, and kilometers.

• Metric units for mass include milligrams, grams, and kilograms.

• Metric units for volume include cubic millimeters, cubic centimeters, and cubic meters.

• Metric units for capacity include milliliters, centiliters, liters, and kiloliters.

mode

The value found most often in a set of numbers. There may be no mode, one mode, or more than one mode in a set of numbers. It is a measure of central tendency.

371

Appendix C: 6-12 Mathematics Glossary

piecewise function

A function defined by multiple sub functions, each of which applies to a certain interval of the main function’s domain.

polynomials

The sum or difference of terms which have variables raised to non-negative integer powers and which have coefficients that may be real or complex.

population (in data analysis)

The entire set of cases or individuals under consideration in a statistical analysis.

prime factorization

The expression of a number as the product of prime factors.

prime number

A whole number greater than 1 that is not divisible by any whole number other than 1 and itself.

principal square roots

The principal square root is the positive square root of a positive real number.

proportional relationships

A collection of pairs of numbers that are in equivalent ratios.

A value that is much higher or much lower than the other values in a set of data.

quadrant

Any of the four regions separated by the axes in a coordinate plane.

percent of change

The difference between a final value and an initial value, expressed as a percentage of the initial value.

quadratic expression (or quadratic equation)

A polynomial expression or equation that contains a term of degree 2, but no term of higher degree.

percent error

The difference between the estimated number and the actual number as a percentage of the actual value.

quadratic function

A polynomial function with degree of 2.

quartiles

For a data set with median M, the first quartile is the median of the data values less than M and the third quartile is the median of the data values greater than M. The second quartile is the median M.

radius

A line segment extending from the center of a circle or sphere to a point on the circle or sphere.

monomial

A polynomial with one term.

multiplicative identity

Refer to Properties of Operations, Equality and Inequality (Appendix D).

multiplicative inverse (reciprocal)

Refer to Properties of Operations, Equality and Inequality (Appendix D).

net

A two-dimensional diagram that can be folded or made into a three-dimensional figure.

order of operations

The order of performing computations is to first work within grouping symbols using the order of operations. Then simplify terms with exponents. Next, while reading from left to right, perform multiplication and division in the order in which it appears. Finally, while reading from left to right, perform addition and subtraction in the order in which it appears.

origin

outlier

pi (𝜋𝜋)

372

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

The point of intersection of the 𝑥𝑥- and 𝑦𝑦-axes in a rectangular coordinate system, where the 𝑥𝑥coordinate and 𝑦𝑦-coordinate are both 0.

The symbol designating the ratio of the circumference of a circle to its diameter. It is an irrational number. Common approximations are 3.14,

!! $

or

%&& ##%

.

Appendix C: 6-12 Mathematics Glossary

random sampling

A smaller group of people or objects chosen from a larger group or population by a process giving equal chance of selection to all possible people or objects, and all possible subsets of the same size.

random variable

An assignment of a numerical value to each outcome in a sample space.

range (of a data set)

The difference between the highest data value and the lowest data value.

range (of a relation or function)

The complete set of possible values of the output of a relation or function. See domain.

rate

A ratio that compares two quantities of different units.

rate of change

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

rigid transformation

A transformation of points in space consisting of a sequence of one or more translations, reflections, or rotations. Rigid transformations preserve distances and angle measures (congruency).

rotation

A transformation of a figure by turning it about a center point or axis. The amount of rotation can be expressed in the number of degrees.

The direction of the rotation for twodimensional figures can be expressed as clockwise or counterclockwise.

sample space

In a probability model for a random process, a list of the individual outcomes that are to be considered.

The ratio of change in one quantity to the corresponding change in another quantity.

scale

The numeric values, set at fixed intervals, assigned to the axes of a graph.

rational expression

A quotient of two polynomials with a non-zero denominator.

scale factor

rational number

A real number that can be expressed as the ratio of two integers.

The constant that is multiplied by the length of each side of a figure to produce an image that is the same shape as the original figure.

scale model

A model or drawing based on a ratio of the dimensions for the model and the actual object it represents.

scatter plot

A graph in the coordinate plane representing a set of bivariate numerical data that is used to observe the relationship between two variables. See line of fit.

scientific notation

A method of writing very large or very small numbers using exponents in which a number is expressed as the product of a power of 10 and a number that is at least 1 and less than 10.

real numbers

The set of all rational and irrational numbers.

reflection

A transformation that produces the mirror image of a geometric figure across a line of reflection.

regular polygon

A polygon that is both equilateral (all sides congruent) and equiangular (all angles congruent).

relation

A set of input-output pairs.

repeated experiment

A random experiment done with the same conditions and parameters as a previous one.

373

Appendix C: 6-12 Mathematics Glossary

set-builder notation

significant digits

similarity

A shorthand used to write sets, often sets with an infinite number of elements. The set {𝑥𝑥: 𝑥𝑥 > 0} is read aloud, "the set of all 𝑥𝑥 such that 𝑥𝑥 is greater than 0." It is read aloud exactly the same way when the colon : is replaced by the vertical line | as in {𝑥𝑥|𝑥𝑥 > 0}.

The nonzero digits of a number and the zeros that are included between them or any trailing zeros that are considered to be precise.

Having exactly the same shape but not necessarily the same size. Equivalently, two figures are similar if one can be mapped to the other using a rigid transformation combined with a dilation, including cases with a scale factor of 1.

simple interest

A method of computing interest. Interest is computed from the (original) principal alone no matter how much money has accrued so far.

simulation

A simulation is an approximate imitation of a statistical experiment, often done with a computer program to examine the statistics of a large quantity of trials.

slope

374

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

The ratio of the change in the vertical direction (𝑦𝑦 direction) to change in the horizontal direction (𝑥𝑥 direction), often expressed as

∆( ∆)

theoretical probability

A number between 0 and 1 representing the likelihood of an event in a theoretical model based on a sample space. If all outcomes in the sample space are equally likely, then theoretical probability of an event is the ratio of the number of outcomes in the event to the number of outcomes in the sample space.

translation

A transformation in which every point in a figure is moved in the same direction and by the same distance.

transversal

A line that intersects two or more lines in the same plane at different points.

trigonometric function

Any of the six functions (sine, cosine, tangent, cotangent, secant, cosecant) that, for an acute angle of a right triangle, may be expressed in terms of ratios of sides of the right triangle.

unit rates

A ratio comparing a number of units of one quantity to one unit of a second quantity.

vertical angles

The opposite angles formed when two lines intersect.

𝑥𝑥-intercept

The value of 𝑥𝑥 at the point where a line or graph intersects the 𝑥𝑥-axis. The value of 𝑦𝑦 is 0 at this point.

𝒚𝒚-intercept

The value of 𝑦𝑦 at the point where a line or graph intersects the 𝑦𝑦-axis. The value of 𝑥𝑥 is 0 at this point.

.

statistical question

A question that can be answered by collecting data. Often there will be variability in the data.

stem-and-leaf plot

A table that organizes data by place value to compare data frequencies.

supplementary angles

Two angles with measures the sum of which is exactly 180°.

Appendix D: Properties of Operations, Equality and Inequality

FLORI DA’ S B. E. S. T. STANDARDS : MATHEMATI CS

Properties of Operations

Properties of Equality

The table below illustrates the properties of operations. For each property, the variables 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system and the complex number system.

Property of Operation Associative property of addition Commutative property of addition Additive identity property of zero Existence of additive inverses Associative property of multiplication Commutative property of multiplication Multiplicative identity property of one Existence of multiplicative inverses Distributive property of multiplication over addition

Property of Equality

Example

Reflexive property of equality

(𝑎𝑎 + 𝑏𝑏) + 𝑐𝑐 = 𝑎𝑎 + (𝑏𝑏 + 𝑐𝑐)

Symmetric property of equality

𝑎𝑎 + 𝑏𝑏 = 𝑏𝑏 + 𝑎𝑎

Transitive property of equality

𝑎𝑎 + 0 = 𝑎𝑎 0 + 𝑎𝑎 = 𝑎𝑎

For every 𝑎𝑎 there exists −𝑎𝑎 so that 𝑎𝑎 + (−𝑎𝑎) = 0 and (−𝑎𝑎) + 𝑎𝑎 = 0. (𝑎𝑎 × 𝑏𝑏) × 𝑐𝑐 = 𝑎𝑎 × (𝑏𝑏 × 𝑐𝑐)

𝑎𝑎 × " = 1 and

Subtraction property of equality

Division property of equality

For every 𝑎𝑎 ≠ 0 there exists !

Addition property of equality

Multiplication property of equality

𝑎𝑎 × 𝑏𝑏 = 𝑏𝑏 × 𝑎𝑎 𝑎𝑎 × 1 = 𝑎𝑎 1 × 𝑎𝑎 = 𝑎𝑎

The table below illustrates the properties of equality. For each property, the variables 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 stand for arbitrary numbers in a given number system. The properties of equality apply to the rational number system, the real number system and the complex number system.

!

"

× 𝑎𝑎 = 1.

𝑎𝑎 × (𝑏𝑏 + 𝑐𝑐) = (𝑎𝑎 × 𝑏𝑏) + (𝑎𝑎 × 𝑐𝑐)

!

"

so that

Substitution property of equality

Example 𝑎𝑎 = 𝑎𝑎

If 𝑎𝑎 = 𝑏𝑏, then 𝑏𝑏 = 𝑎𝑎.

If 𝑎𝑎 = 𝑏𝑏 and 𝑏𝑏 = 𝑐𝑐, then 𝑎𝑎 = 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 + 𝑐𝑐 = 𝑏𝑏 + 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 − 𝑐𝑐 = 𝑏𝑏 − 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 × 𝑐𝑐 = 𝑏𝑏 × 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏 and 𝑐𝑐 ≠ 0, then 𝑎𝑎 ÷ 𝑐𝑐 = 𝑏𝑏 ÷ 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏, then 𝑏𝑏 may be substituted for 𝑎𝑎 in any expression containing 𝑎𝑎.

375

Appendix D: Properties of Operations, Equality and Inequality

FLORI DA’ S B. E. S. T. STANDARDS : MATHEMATI CS

Properties of Inequality The table below illustrates the properties of inequality. For each property, the variables 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 stand for arbitrary numbers in a given number system. In addition, exactly one of the following is true: 𝑎𝑎 < 𝑏𝑏, 𝑎𝑎 = 𝑏𝑏, or 𝑎𝑎 > 𝑏𝑏. The properties of inequality apply to the rational number system and the real number system.

Property of Inequality Asymmetric property of inequality Transitive property of inequality Addition property of inequality Subtraction property of inequality Multiplication property of inequality Division property of inequality

376

Example If 𝑎𝑎 > 𝑏𝑏, then 𝑏𝑏 < 𝑎𝑎.

If 𝑎𝑎 > 𝑏𝑏 and 𝑏𝑏 > 𝑐𝑐, then 𝑎𝑎 > 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏, then 𝑎𝑎 + 𝑐𝑐 > 𝑏𝑏 + 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏, then 𝑎𝑎 − 𝑐𝑐 > 𝑏𝑏 − 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 > 0, then 𝑎𝑎 × 𝑐𝑐 > 𝑏𝑏 × 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 < 0, then 𝑎𝑎 × 𝑐𝑐 < 𝑏𝑏 × 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 > 0, then 𝑎𝑎 ÷ 𝑐𝑐 > 𝑏𝑏 ÷ 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 < 0, then 𝑎𝑎 ÷ 𝑐𝑐 < 𝑏𝑏 ÷ 𝑐𝑐.

Appendix E: K-12 Formulas

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

Area of a two-dimensional figure 𝐴𝐴 = 𝑙𝑙𝑙𝑙, where 𝑙𝑙 is the length and 𝑤𝑤 is the width 𝐴𝐴 = 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height

Rectangle

𝐴𝐴 = 𝑙𝑙𝑙𝑙, where 𝑙𝑙 is the length and 𝑤𝑤 is the width 𝐴𝐴 = 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height 𝐴𝐴 = 𝑠𝑠 ! , where 𝑠𝑠 is the side length

Square

"

𝐴𝐴 = ! 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height

Triangle

"

Trapezoid Parallelogram

𝐴𝐴 = ! (𝑏𝑏" + 𝑏𝑏! )ℎ, where 𝑏𝑏" and 𝑏𝑏! are the bases and ℎ is the height

𝐴𝐴 = 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height 𝐴𝐴 = 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height " 𝐴𝐴 = 𝑑𝑑" 𝑑𝑑! , where 𝑑𝑑" and 𝑑𝑑! are the diagonals

Rhombus

!

𝐴𝐴 = 𝜋𝜋𝑟𝑟2 , where 𝑟𝑟 is the radius

Circle Equilateral triangle Regular polygon

𝐴𝐴 =

√% ! 𝑠𝑠 , & "

where 𝑠𝑠 is the side length

𝐴𝐴 = ! 𝑃𝑃𝑃𝑃, where 𝑃𝑃 is the perimeter and 𝑎𝑎 is the apothem

Surface Area of a three-dimensional figure Cube Prism Cylinder Cone Pyramid Regular pyramid Sphere

𝑆𝑆𝑆𝑆 = 6𝑠𝑠 , where 𝑠𝑠 is the side length !

𝑆𝑆𝑆𝑆 = 2𝐵𝐵 + 𝑃𝑃ℎ, where 𝐵𝐵 is the area of the base, 𝑃𝑃 is the perimeter of the base and ℎ is the height 𝑆𝑆𝑆𝑆 = 2𝐵𝐵 + 𝑃𝑃ℎ, where 𝐵𝐵 is the area of the base, 𝑃𝑃 is the perimeter of the base and ℎ is the height

𝑆𝑆𝑆𝑆 = 𝐵𝐵 + 𝜋𝜋𝜋𝜋ℎ' , where 𝐵𝐵 is the area of the base, 𝑟𝑟 is the radius and ℎ' is the slant height 𝑆𝑆𝑆𝑆 = 𝐵𝐵 + 𝐴𝐴(𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓), where 𝐵𝐵 is the area of the base and 𝐴𝐴(𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓) is the area of each face "

𝑆𝑆𝑆𝑆 = 𝐵𝐵 + ! 𝑃𝑃ℎ' , where 𝐵𝐵 is the area of the base, 𝑃𝑃 is the perimeter of the base and ℎ' is the slant height

𝑆𝑆𝑆𝑆 = 4𝜋𝜋𝑟𝑟 ! , where 𝑟𝑟 is the radius

𝑆𝑆𝑆𝑆 = 3𝜋𝜋𝑟𝑟 ! , where 𝑟𝑟 is the radius and the area of the flat side is Hemisphere included

Volume of a three-dimensional figure Cube Prism Cylinder Cone Pyramid Sphere

𝑉𝑉 = 𝑠𝑠 ! , where 𝑠𝑠 is the side length

𝑉𝑉 = 𝐵𝐵ℎ, where 𝐵𝐵 is the area of the base and ℎ is the height 𝑉𝑉 = 𝐵𝐵ℎ, where 𝐵𝐵 is the area of the base and ℎ is the height "

𝑉𝑉 = ! 𝐵𝐵ℎ, where 𝐵𝐵 is the area of the base and ℎ is the height " 𝑉𝑉 = 𝐵𝐵ℎ, where 𝐵𝐵 is the area of the base and ℎ is the ! height #

𝑉𝑉 = ! 𝜋𝜋𝑟𝑟 ! , where 𝑟𝑟 is the radius

Laws of Exponents (where 𝑚𝑚 and 𝑛𝑛 are integers and 𝑎𝑎 and 𝑏𝑏 are nonzero real numbers)

Product of powers Quotient of powers Power of a power Power of a product Power of a quotient Negative exponent Identity exponent Zero exponent

𝑎𝑎! ⋅ 𝑎𝑎" = 𝑎𝑎!#" and conversely 𝑎𝑎()* = 𝑎𝑎( ⋅ 𝑎𝑎* !! !"

!!

= 𝑎𝑎"#$ and conversely 𝑎𝑎"#$ =

!"

(𝑎𝑎" )$ = 𝑎𝑎"⋅$ and conversely 𝑎𝑎"⋅$ = (𝑎𝑎" )$

(𝑎𝑎𝑎𝑎)( = 𝑎𝑎( ⋅ 𝑏𝑏( and conversely 𝑎𝑎( ⋅ 𝑏𝑏( = (𝑎𝑎𝑎𝑎)( ! "

& ' &

𝑎𝑎

+"

&

"

𝑎𝑎" = 𝑎𝑎

=

𝑎𝑎- = 1 !

!!

and conversely

&!

= , and conversely

! #'

& '

=

"

&

!

!

and conversely

"

𝑎𝑎 " = ( √𝑎𝑎*

Rational, fractional where 𝑎𝑎 > 0 ! " exponent " "

𝑎𝑎

'

!! &!

= 𝑎𝑎 &

!

! "

= & '

#'

&

! #'

= & ' &

"

"

and conversely ( √𝑎𝑎* "

!

= 𝑎𝑎 " ,

!

= ( √𝑎𝑎 * and conversely ( √𝑎𝑎" * = 𝑎𝑎 " ,

where 𝑎𝑎 > 0

377

NOTES

378

NOTES

379

NOTES

380

NOTES

381

NOTES

382

NOTES

383

384

G211