MoreCircles

Circles Rolando & Nicola edition

Lines and Segments Circle in Coordinate Planes

Arc measures Angle and Segment Relationships

Chords

& Inscribed Angles and Polygons

Table of Contents Page 4 - Lines & Segments Page 5 & 6 - Coplanar Circles and Common Tangents Page 7 - Arc Measures Page 8 - Congruent Circles and Arc Page 9 - Fun Crossword Puzzle Page 10 - Chords Page 11 - Inscribed Angles Page 12 - Inscribed Polygons Page 13 & 14 - Finding Angle and Arc Measures Page 15 & 16 - Using Segments of Chord, Tangents, and Secants Page 17 - Circles in Coordinate Planes Page 19 - Do It Yourself Page 20 - References

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Lines & Segments A segment whose endpoints are the center and point on a circle is a radius.

A chord is a segment whose endpoints are on a circle.

A diameter is a chord that contains the center of the circle.

A secant is a line that intersects a circle in two points.

A tangent is a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency.

Coplanar Circles and Common Tangents Coplanar circles that intersect in one point are called tangent circles.

Coplanar circles that have a common center are called concentric circles.

A line or segment that is tangent to two coplanar circles is called a common tangent . Common internal tangent intersects the segment that joins the center of two circles. A common external tangent does not intersect the segment that joins the center of two circles.

Theorems Tangent Line to Circle Theorem a line is tangent to a circle only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

Example

External Tangent Congruence Theorem Tangent segments from a common external point are congruent

How many common tangents are there between Earth and the Moon. Green for External and Red for Internal common tangents.

Arc Measures A central angle of a circle is an angle whose vertex is the center of the circle.

Minor arcs are less than half of a rotation (180°), and they are represented by their two endpoints. In the example below (AB.

Major Arc is the bigger part of the circle being the two endpoints of the minor arc and another point in the circle. In the example below (ABC.

The measure of a Minor Arc is the measure of the central angle.

The measure of a Major Arc is the minor arc measure minus 360°.

Congruent Circles and Arc Two circles are congruent circles only if a rigid motion or a composition of rigid motions maps one circle onto the other.

Two arcs are congruent arcs only if they Have the same measure and they are arcs a the same circle ar of congruent circle.

Examples T

90° S

P

Find -TS -FTS -FST

F Solution TS is a minor arc so its 90 degrees FTS is a major arc and TS is 90 degrees so FTS is 360-90=270 degrees FT is a diameter, so FST is a semicircle, FST=180 degrees

Crossword Puzzle

Vocabulary from Lines and Segments

Chords Theorems Congruent corresponding Chords Theorem In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Perpendicular Chord Bisector Theorem If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord.

Perpendicular Chord Bisector Converse If one chord of a circle is a perpendicular bisector of another chord, then the first chord

Example

Find FG

Solution

Because FG and JK are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent by the Congruent Corresponding Chords Theorem.

So FG = K = 80

Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

Intercepted Arc

An arc that lies between two lines, rays, or segments is called an intercepted arc. If the endpoints of a chord or arc lie on the sides of an inscribed angle, then the chord or arc is said to subtend the angle.

Inscribed Polygon A polygon is an inscribed polygon when all angles vertices lie on a circle. The circle that contains the vertices is a circumscribed circle.

Using Inscribed Angles

Using Inscribed Polygons

Finding Angle and Arc measures

Circumscribed Angle : A circumscribed angle is an angle whose sides are tangent to a circle.

Theorems you need to know Tangent and Intersecting Chord Theorem- If a tangent and a chord intersect at a point on a circle. then the measure of each angle formed is onehalf the measure of its intercepted arc.

Angles Inside the Circle Theorem- If two chords intersect inside a circle, then the measure of each angle is onehalf the core of the measures of the ares intercepted by the angle and its vertical angle.

Angles Outside the Circle Theorem-If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures the intercepted ares.

Circumscribed Angle Theorem -The measure of a circumscribed angle is equal to 180 minus the measure of the central angle that intercepts the same arc

e pl am Ex

Use the circumscribed angle theorem and solve

A D

135

X

B

ADB= 180 - ACB

ADB= 180 - 135

X= 45

D

Using Segments of Chord, Tangents, and Secants Tangent Segment and Secant Segment:

A secant segment is a segment that contains a chord of a circle and has exactly one endpoint outside the circle.

The part of a secant segment that is outside the circle is called an external segment.

Segments of Chords Theorem If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

Segments of Secants Theorem If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and is external segment equals the product of the lengths of the other secant segment and its external segment.

Theorems

A tangent segment is a segment that is tangent to a circle at an endpoint.

Segments of Secants and Tangents Theorem If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

Example Find the value of X

t

8

q

s

X 16

r

Circles in Coordinate Planes Standard Equation of a Circle: (x, y) represent any paint on a circle with center (h, k) and radius r. By the Pythagorean Theorem

(x-h)^2+ (y-k)^2 = r^2 This is the standard equation of a circle with center (h,k) and radius r

Example

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Do It Yourself 1- In the Advertisement on the page above, can you draw lines and segments and name them.

2- Graph the circle. (x-3)^2 + (y+1)^2 = 25

References -Illustration page 10, Big Ideas Math common core high school geometry page 546 ~Ron Larson and Laurie Boswell -Illustration page 12, Big Ideas Math common core high school geometry page 556 ~Ron Larson and Laurie Boswell -Illustration page 16, Big Ideas Math common core high school geometry page 572 ~Ron Larson and Laurie Boswell

-Big Ideas Math common core high school geometry Chapter 10 ~Ron Larson and Laurie Boswell

Circles by: Rolando and Nicola

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