Grade 10 Facilitator's Guide 2/2 Mathematics

Facilitator’s Guide 2/2

10

Mathematics

CAPS

Mathematics FET Phase Grade 10 • Facilitator’s Guide 2/2 home classroom college workplace

2210-E-MAM-FG02

9 781990

948497

Mathematics Facilitator's Guide 2/2 – Grade 10

2210-E-MAM-FG02

9 781990

948497

CAPS-aligned

Prof. C Vermeulen, Lead author P de Swardt H Otto M Sherman E van Heerden L Young

10.1 Interest...........................................................................................................51

TABLE OF CONTENTS

Exercise 10.1: Quiz.................................................................................................... 52

10.2 Simple interest.............................................................................................53 PREFACE..................................................................................................................... 1

Exercise 10.2................................................................................................................ 55

THEME 9: ANALYTICAL GEOMETRY......................................................................... 7

10.3 Hire purchase...............................................................................................58

Introduction....................................................................................................................7

Exercise 10.3................................................................................................................ 61

Revision exercise...........................................................................................................9

Self evaluation............................................................................................................. 66

9.1

The distance formula.................................................................................10

10.4 Compound interest.....................................................................................67

9.2

Exercise 9.1: The distance formula.................................................................... 12

Exercise 10.4................................................................................................................ 69

The mid-point of a line segment............................................................19

10.5 Timelines.......................................................................................................72

Exercise 9.2: The mid-point of a line segment.............................................. 20

Exercise 10.5................................................................................................................ 75

Self-evaluation............................................................................................................ 26

Self evaluation............................................................................................................. 80

9.3

The gradient (inclination) of a line.......................................................26

10.6 Comparing financial situations..............................................................81

9.4

Exercise 9.3: The gradient (inclination) of a line......................................... 28

Points on a straight line (collinear points).........................................36

10.7 Inflation and population growth............................................................85

9.5

Exercise 9.4: Points on a straight line (collinear points).......................... 37

Exercise 10.6................................................................................................................ 83 Exercise 10.7................................................................................................................ 89

The equation of a straight line ...............................................................40

10.8 Foreign exchange rates..............................................................................93

Exercise 9.5: The equation of a straight line.................................................. 40

Exercise 10.8................................................................................................................ 95

Self evaluation............................................................................................................. 43

Self evaluation.......................................................................................................... 100

Summary of theme.................................................................................................... 44 End of theme exercise.............................................................................................. 44

Summary of theme................................................................................................. 100

THEME 10: FINANCE AND GROWTH.....................................................................49

THEME 11: STATISTICS.........................................................................................108

Introduction................................................................................................................. 49

Introduction.............................................................................................................. 108

End of theme exercise........................................................................................... 101

Revision exercise........................................................................................................ 50

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Revision exercise..................................................................................................... 109 i

G10 – Mathematics

G10 – Mathematics – Facilitator's Guide 2/2 11.1 Grouped data.............................................................................................113

THEME 12: TRIGONOMETRY: PROBLEMS IN TWO DIMENSIONS...................164

Exercise 11.1: Grouped data.............................................................................. 116

Introduction.............................................................................................................. 164

11.2 Measures of central tendency in grouped data.............................. 122

Revision exercise..................................................................................................... 164

11.2.1 Estimated mean of grouped data.......................................................122

12.1 Problems involving angles of elevation and depression............. 166

11.2.2 Estimated median of grouped data................................................... 123

Exercise 12.1: Problems involving angles of elevation and depression......................................................................................................... 169

11.2.3 Modal class of grouped data................................................................123

12.2 Problems involving bearings (Direction)......................................... 173

Exercise 11.2: Measures of central tendency in grouped data............ 126

Exercise 12.2: Problems involving bearings (direction)........................ 175

Self-evaluation......................................................................................................... 128

Self-evaluation......................................................................................................... 180

11.3 Measures of spread..................................................................................129 11.3.2 Interquartile range (IQR)......................................................................130

Summary of theme................................................................................................. 180

11.3.3 Semi-interquartile range ......................................................................130

THEME 13: EUCLIDEAN GEOMETRY................................................................... 184

11.3.4 Percentiles..................................................................................................130

Introduction.............................................................................................................. 184

11.3.1 Quartiles......................................................................................................129

End of theme exercise........................................................................................... 180 Exercise 13.1............................................................................................................. 185

11.3.5 Outliers........................................................................................................130

Self-evaluation......................................................................................................... 198

Exercise 11.3: Measures of spread.................................................................. 136

Summary of theme................................................................................................. 198

11.4 The five-number summary...................................................................139

Exercise 11.4: The five-number summary................................................... 140

THEME 14: MEASUREMENT.................................................................................199

Exercise 11.5: Analysing statistical summaries and graphs................. 146

14.1 Revision: Perimeter, area, surface area and volume.................... 203

End of theme exercise........................................................................................... 150

14.1.3 Nets of 3D shapes.....................................................................................203

Introduction.............................................................................................................. 199

Self-evaluation......................................................................................................... 143

11.5 Analysing statistical summaries and graphs .................................. 144

Revision exercise..................................................................................................... 201

14.1.1 Perimeter/circumference and area................................................... 203

Self-evaluation......................................................................................................... 149

14.1.2 Regular polygons......................................................................................203

Summary of theme................................................................................................. 150

14.1.4 Surface area and volume........................................................................204

ii

Exercise 14.1: Revision: Perimeter, area, surface area and volume.. 204

14.2

Surface area and volume of cones......................................................216

15.2.2 Definition of relative frequency..........................................................250

Exercise 14.2: Surface area and volume of cones..................................... 217

Exercise 15.2: Relative frequency.................................................................... 251

14.3 Surface area and volume of pyramids...............................................220

Self-evaluation......................................................................................................... 251

Exercise 14.3: Surface area and volume of pyramids.............................. 222

15.3 Venn diagrams...........................................................................................252

14.4 Surface area and volume of spheres.................................................. 228

15.3.1 Using Venn diagrams to represent events........................................ 252

Exercise 14.4: Surface area and volume of spheres................................. 230

15.3.2 Union and intersection...........................................................................252

Self-evaluation......................................................................................................... 232

15.3.3 Using Venn diagrams to determine probability............................. 255

multiplied by a constant factor ​k.​....................................................... 233

15.4 Probability identities..............................................................................261

Exercise 14.5: Effect on volume and surface area when any dimension is multiplied by a constant factor k................................................................. 234

15.4.1 The union of two events.........................................................................261

14.5 Effect on volume and surface area when any dimension is

Exercise 15.3: Venn diagrams............................................................................ 257

15.4.2 Mutually exclusive events......................................................................263

Self-evaluation......................................................................................................... 236

15.4.3 Complementary events...........................................................................264

Summary of theme................................................................................................. 237

15.4.4 Mutually exhaustive events...................................................................265

End of theme exercise........................................................................................... 237

Exercise 15.4: Probability identities, mutually exclusive events and complementary events......................................................................................... 266

THEME 15: PROBABILITY.....................................................................................242

Introduction.............................................................................................................. 242

Self-evaluation......................................................................................................... 268

15.1 Theoretical probability..........................................................................246

Summary of theme................................................................................................. 268

15.1.1 Terminology...............................................................................................246

PAPERS AND MEMOS...........................................................................................277

15.1.2 Expressing probability...........................................................................246

FORMULAE..............................................................................................................304

Revision exercise..................................................................................................... 243

End of theme exercise........................................................................................... 269

15.1.3 Definition of probability........................................................................247

EUCLIDIAN GEOMETRY: ACCEPTED REASONS..................................................307

Exercise 15.1: Theoretical probability........................................................... 248

INDEX......................................................................................................................310

15.2 Relative frequency...................................................................................249 15.2.1 The difference between relative frequency and theoretical probability...........................................................................249

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iii

G10 – Mathematics

PREFACE

Term

In Grade 10, mathematics is an optional subject (as an alternative to mathematical literacy) for the first time. There may be various reasons why you might choose mathematics as a subject, for example to prepare you for a field of study where Grade 12 mathematics is a prerequisite, or a career in which a background in mathematics would be advantageous.

1

(11 weeks)

In general, mathematics in Grade 10 to 12 involves more abstract concepts and more complex procedures than in Grade 1 to 9. Mastering mathematics in Grade 10 to 12 requires more time, commitment, critical thought and reflection than in Grade 1 to 9.

2

This product consists of two study guides and two facilitator’s guides, which are based on the concepts of Optimi’s GuidEd Learning™ model to help you achieve success in your study of mathematics. These books cover all work required for Grade 10 mathematics and have been compiled in accordance with the CAPS guidelines as required by the Department of Basic Education.

(11 weeks)

Number of weeks 3

1. Algebraic expressions

2

3. Number patterns

2 2

3 4

1 3 3 2

The study guides are supported by supplementary lesson structures on the Optimi Learning Platform (OLP), which is an online platform. These lesson structures offer continuous guidance to support and enrich your learning process. This guidance is based on the latest insights in education, cognitive psychology and neuroscience. Note that the study guides can also be used independently of the OLP.

3

(10 weeks)

The study guides and facilitator’s guides are divided into 15 themes. Study guide 1/2 and facilitator’s guide 1/2 cover themes 1 to 8 (terms 1 and 2) and study guide 2/2 and facilitator’s guide 2/2 cover themes 9 to 15 (terms 3 and 4).

The themes correspond with the CAPS guidelines with regard to content and time allocation and represent the year plan.

2 2 2

1 1

4

(10 weeks)

Theme

2

4 4

2. Exponents

4. Equations and inequalities 5. Trigonometry 6. Functions

7. Trigonometric functions 8. Euclidean geometry

Assessment* Investigation or project Test

Assignment or test

Practice papers are included in the textbook.

June examination

10. Finance and growth

Test

9. Analytical geometry 11. Statistics

12. Problems in two dimensions 13. Euclidean geometry 14. Measurement 15. Probability Revision

Practice papers are included in the textbook.

Test Test End-of-year examination

* You will find the latest and most comprehensive information on assessment in the portfolio book and assessment plan. © Optimi

1

G10 – Mathematics

G10 – Mathematics – Facilitator's Guide 2/2

Time allocation

Each theme has the same structure in order to make it easier for you to navigate through them. The structure is as follows:

According to the CAPS requirements, at least 4,5 hours should be spent on teaching mathematics per week. For example, 13,5 hours (three weeks × 4,5 hours per week) will be spent on teaching Theme 1 (algebraic expressions).

Themes have not been sub-divided into lessons; you and your facilitator are at liberty to complete as much content per session and per week as your progress allows. If you work at a slower pace, the necessary adaptations should be done so that you will still be able to master all the work in time.

Introduction

What this theme is about This briefly tells you what the theme is about without providing details or using “difficult” or unknown concepts. A comprehensive list of the learning outcomes you need to master in a specific theme is given as a summary at the end of the theme.

Note that the teaching time referred to above does not include the time during which you should apply and practise the knowledge and concepts you have learned. For this purpose, various exercises are provided throughout each theme. These exercises involve different ways of applying and practising new knowledge and cover various degrees of difficulty. You should try to do all of these exercises. Complete solutions are provided in the facilitator’s guide.

Previous knowledge

This section tells you what existing knowledge you need to master the theme involved.

The learning activities available in the OLP’s lesson structures involve different formats and levels of interaction. The resources not only support the learning process, but also offer you the opportunity to practise new knowledge.

Tip: The more e𝑥ercises you do, the greater the chance that you will achieve success in mathematics.

Structure of themes Learning is a complex process. Millions of brain cells and neural pathways in our brains work together to store new information in the long-term memory so that we will be able to remember it later on. Long-term memory is not our only type of memory and when we learn, our working memory is just as important. Working memory is different from long-term memory and has a limited capacity. This means that one’s working memory can only handle a small amount of new information at a time.

When one learns mathematics, there is a lot of new information your brain needs to process, which can easily exhaust your working memory. The study guides have been written and compiled in such a way that it does not overtax the working memory and therefore simplifies the process of learning mathematics.

2

Revision

Exercises

This may involve one of the following: 1.

2.

3.

The exercises give you the opportunity to practise the concepts and procedures taught. It is important to try and complete all exercises. Complete solutions are provided in the facilitator’s guides.

revision of the concepts, definitions and procedures required as previous knowledge, an exercise or activity with solutions so that you can test your previous knowledge yourself, or a combination of the above.

Questions usually progress from easy (in order to master and practise basic concepts and procedures) to difficult (more complex operations).

Mixed exercises are also provided, where you get the opportunity to practise different concepts and procedures and integrate these with previous themes.

Do not neglect this revision. It is important to work through this section thoroughly. Mathematical concepts often follow on one another and if basic knowledge is lacking or has not been mastered sufficiently, this will handicap the formation of new knowledge.

Summary of theme

Following the introductory part of the theme, new knowledge is dealt with in sub-themes. Each sub-theme has the following structure:

Here you will find a summary of what you should have mastered in the theme.

This is expressed in more formal mathematical language in order to be in keeping with the CAPS (the curriculum statement).

SUB-THEME

Introduction

End of theme exercise

New concepts and procedures are explained. Relevant previous knowledge is also dealt with here if necessary.

This is a mixed exercise involving all concepts and procedures dealt with in the theme, where this work can also be integrated with previous work. The degree of difficulty of this exercise varies. It is important that you try and complete all the exercises. Complete solutions can be found in the facilitator’s guide.

Worked examples

Mixed exercises such as these in this textbook form a very important component of mastering mathematics. There is a big difference between the ability to recognise your work and the ability to recall it. When you are able to recognise your work, you will often say “Oh, of course!” but you struggle to remember this when you are writing an examination. When you are able to recall your work, this means that you have captured that knowledge in your long-term

Worked examples show you how the new concepts and procedures are applied and help you understand and apply the newly taught concepts and procedures.

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3

G10 – Mathematics

G10 – Mathematics – Facilitator's Guide 2/2

memory and are able to remember and use it. Mixed exercises enable you to not only recognise the work, but also recall it from your long-term memory.

Self-evaluation

When you practise the same type of sum or problem over and over, you often get lazy and do not reflect upon the exercise anymore. You are convinced that you know exactly what type of sum or problem you need to solve. But in a test or exam, all these problems are mixed up and then it might be difficult to know what to do. When mixed exercises form part of your learning process, you learn to identify and complete a sum or problem correctly. This means that you are truly prepared for tests or exams, because you can recall your work instead of merely recognising it.

In each theme, and usually following each sub-theme, there is an activity where you need to reflect critically about the extent to which you have mastered certain concepts and procedures. This activity has the following format:

Use the following scale to determine how comfortable you are with each topic in the table below:

1. Help! I don’t feel comfortable with the topic at all. I need help.

2. Alarm! I don’t feel comfortable, but I just need more time to work through the topic again.

3. OK! I feel moderately comfortable with the topic, but I still struggle sometimes.

4. Sharp! I feel comfortable with the topic.

5. Whoo-hoo, it’s party time! I feel totally comfortable with the topic and can even answer more complicated questions about it.

Complete the table:

Topic

1

2

3

4

5

Tip: Complete each self-evaluation as honestly as possible. If there are aspects which you have not mastered, revisit these and make sure that you do master them. Ask the facilitator for help. It is important not to move on to a next theme or sub-theme before you have mastered the topic involved, even if this means that you spend more time on a specific theme than recommended by the CAPS.

4

Exam papers

Supplementary books

Visit Impaq’s online platform for the assessment plan and comprehensive information about the compilation and mark allocation of tests, assignments and examinations.

Any other books can be used along with this textbook for extra exercises and explanations, including: • Maths 4 Africa, available at www.maths4africa.co.za • The Siyavula textbook, available online for free at www.siyavula.com • Pythagoras, available at www.fisichem.co.za.

Tip: Make sure that you know which themes are covered in which paper. The themes covered in the e𝑥amination papers are subject to change. Always refer to the portfolio book and assessment plan for updated information about the composition of the e𝑥amination papers.

Paper 1

Tip: Use supplementary resources for further e𝑥planations, e𝑥amples, and especially e𝑥tra e𝑥ercises.

Paper 2

Calculator

Algebraic expressions, equations and Euclidean geometry and inequalities, exponents (Theme 1, 2 measurement (Theme 8, 13 and 14) and 4) Analytical geometry (Theme 9) Number patterns (Theme 3) Trigonometry (Theme 5 and 7) Functions and graphs (Theme 6) Statistics (Theme 11) Finance and growth (Theme 10)

We recommend the CASIO fx-82ES (Plus) or CASIO fx-82ZA. However, any scientific, non-programmable and non-graphing calculator is suitable.

Probability (Theme 15)

Note:

• No graphing or programmable calculators are allowed (for example to factorise or find the roots of equations). Calculators should only be used to do standard numeric calculations and to verify calculations done by hand.

• Formula sheets are not provided during tests and final exams in Grade 10.

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5

G10 – Mathematics

THEME 9

Introduction

ANALYTICAL GEOMETRY

In this theme learners will learn to: • • • • •

find the distance between two points in the Cartesian plane find the mid-point of a line segment find the gradient of a line find the equation of a line find the equations of parallel and perpendicular lines and determine whether lines are parallel or perpendicular • work with collinear points (points that lie on the same line) • apply all this knowledge in mixed exercises.

Learning requirements according to CAPS Learners should be able to: 1.

2.

represent geometric figures in a Cartesian coordinate system derive and apply formulae for calculating the following for any two points (𝑥1; y1) and (𝑥2; y2):

• the distance between the two points • the gradient of the line segment connecting the two points (and should be able to determine whether lines are parallel or perpendicular based on their gradients) • the coordinates of the mid-point of the line segment connecting the two points.

Term

Duration Paper Weight

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Prior knowledge

In order to master this theme, learners should already:

• know about the Cartesian plane and that it consists of four quadrants • know about coordinate points, e.g. (3; 5), where the first coordinate (3) is always the 𝑥-coordinate, and the second coordinate (5) is always the y-coordinate • know that the gradient or inclination of a line refers to the steepness of the line, and that it can be defined as:

3

y2 – y1 the change in the y-values between 2 points on the line ___________ ​​ the change in the 𝑥-values between the same 2 points on the line

​​ = ​ _

​​ 𝑥2 – 𝑥1

2 weeks 2

• know the properties of triangles and quadrilaterals.

​15 ± 3 ​of Paper 2

7

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Facilitator's Guide 2/2

What is analytical geometry about?

In the following figure, the point ​​​(​​0; 0​)​​​​has been indicated, which is the origin.

Analytical geometry is also called coordinate geometry and was previously known as Cartesian geometry.

The point ​P​(​​3; 5​)​​​​is also given.

It is the study of geometry using the principles of algebra and the Cartesian coordinate system. It involves the definition of geometric figures in a numerical way (using numbers) and finding numerical information when diagrams are given in the Cartesian plane.

10

II

The development of analytical geometry is sometimes considered the origin of modern mathematics.

–10

Background

If we have the coordinates of the vertices of a figure, we can draw the figure in the Cartesian plane.

III

In the diagram below, for e𝑥ample, ​​L(​ ​​− 5; − 2)​ ​​​​, ​​M(​ ​​− 1; − 6)​ ​​​​ and K ​​ (​ ​​5; 4​)​​​​ are the vertices of ∆ ​ KLM​in the Cartesian plane.

y

5

–5

P(3 ; 5)

I x

5

–5 –10

(0 ; 0) origin

10

IV

The gradient (inclination) of the line from (0;0) to (3;5)

the change in the y-values between 2 points on the line ​= ​___________________________________________________________________ the change in the 𝑥-values between the same 2 points on the line

​ ​y​ ​​ − ​y​ ​​

​= ​_ ​𝑥2​ ​​ − ​𝑥1​ ​​ ​ 2

1

5 − 0 ​= ​_ 3 − 0 ​

​= ​_ 53 ​

8

Let us calculate the gradient of the line in the following figure:

Revision exercise Consider the figure and answer the questions that follow.

the change in the y-values between 2 points on a line

​Gradient = ​___________________________________________________________________ the change in the 𝑥-values between the same 2 points on the line

​ ​y​ ​​ − ​y​ ​​

​= ​_ ​𝑥2​ ​​ − ​𝑥1​ ​​ ​

1.

Write down the coordinates of A, B, C and D.

​= ​_ − 84 ​ ​= − _ ​12 ​

4.

What type of figure is ABCD? Give a reason for your answer.

2

1

2.

1 − 3 ​= ​_ − 8 − 0

​

3.

Calculate the gradients of AD and BC. What do you notice?

Calculate the gradients of AB and DC. What do you notice?

REMEMBER

The order of the letters by which we name a figure is important.

The name ABCD tells us that we move from point A to point B, from B to point C, from C to point D and then back to point A in order to form figure ABCD.

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9

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Facilitator's Guide 2/2 1.

A(–1; 3), B(2; –1), C(8; 1), D(5; 5)

Gradient of BC = ​​_

​​= _ ​​26 ​​= _ ​​ 13 ​​ 8 – 2

2.

3.

4.

5 – 3 Gradient of AD = ​​ _

​​= _ ​​2 ​​= _ ​​1 ​​​​​​​​ 5 – ( – 1) 6 3 1 – ( – 1)

The gradients (inclinations) are equal.

– 1 – 3 Gradient of AB = ​​_ 2 – ( – 1)

​​ = _ ​​ –4 ​​​​ ​​ 3

Gradient of DC = _ ​​ 1 – 5 ​​ = _ ​​–4 ​​ ​​​​ 3 8 – 5

The gradients (inclinations) are equal.

The length of the vertical side is (3 –(–1)) = 4 units and the length of the horizontal side is (2 – (–3)) = 5 units.

It is a parallelogram, because both pairs of opposite sides are parallel.

(distance)² = (3 –(–1))² + (2 – ( – 3))²

9.1 THE DISTANCE FORMULA

(Pythagoras)

= 4² + 5²

= 16 + 25

In the diagram below, two points are given: (–3; –1) and (2; 3). We want to find the distance between them.

= 41

_

In order to find the distance between the points (–3; –1) and (2; 3), we will add a point vertically below (2; 3) and horizontally to the right of (–3; –1). This new point is therefore (2; –1). If we connect the three points, they form a rightangled triangle with the 90° angle at the new point.

​∴​ distance = ​√ 41 ​

We can generalise this calculation and get a formula – a formula is simply the generalisation of a specific case.

We use the theorem of Pythagoras to calculate the distance between (–3; –1) and (2; 3).

REMEMBER

The formula used for calculating the distance between two points, A (​​𝑥​ 1​​​; y1) and B (​𝑥​2; y2), is:

___________________

2 d​​(AB)​​= √ ​​ ​​(​𝑥​ 2​​ – ​𝑥​ 1​​)​​​ 2​ + ​​

(​𝑦​ 2​​ – ​𝑦​ 1​​)​​​ ​ ​​

10

Worked example 1

Worked example 2 The distance between A(​​𝑥; 4​)​​​​and B(2; 1) is 5 units. Find the value of ​𝑥.​

Solution ________________________

AB = ​​√ (𝑥2– 𝑥1 )2+ (𝑦2– 𝑦1 )2 ​​ ___________________

5=√ ​​ (2 – 𝑥)2 + (1 – 4)2 ​​ _________________

5=√ ​​ 4 – 4𝑥 + 𝑥²+ 9 ​​ ______________

5=√ ​​ 𝑥²– 4𝑥 + 13 ​​

25 = 𝑥²– 4𝑥 + 13 0 = 𝑥2 – 4𝑥 – 12

0 = (𝑥 – 6)(𝑥 + 2)

Find the distance between A and B.

𝑥 = 6 or 𝑥 = –2

Solution

The coordinates of A can therefore be (6; 4) or (–2; 4).

Let A(–5 ; –2) = (𝑥1 ; 𝑦1) and B(3 ; 4) = (𝑥2 ; 𝑦2) AB

There are two answers, because there are two possible positions for A. This is shown in the sketch below.

__________________________

=√ ​​ ​​(𝑥2 – 𝑥1 )​​​ 2​ + ​​(𝑦2– 𝑦1 )​​​ 2​ ​​ _________________________

= ​​√ (3–(–5))2 + (4–(–2))2 ​​ _____________

=√ ​​ (8)2 + (6)2 ​​ _

=√ ​​ 100 ​​

= 10 units

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11

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Facilitator's Guide 2/2

Important

2.

Before you start solving a problem in analytical geometry, first draw a sketch showing the information given. Such a sketch helps you to use the information correctly and also to ensure that your answer is correct.

y-axis

C(–4 ; 1) = (𝑥𝑥1 ; y1)

Exercise 9.1: The distance formula

1.

Draw the points C(–4 ; 1) and D(3; –4) on a system of axes and calculate the distance from C to D.

Draw the points A(–1 ; 4) and B(7; –2) on a system of axes and calculate the distance from A to B.

A(–1 ; 4) = (𝑥𝑥1 ; y1)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

y-axis

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

_______________________

CD = √ ​​ ​​(​𝑥​ 2 ​​− ​𝑥​ 1 ​​)​​​ 2​+ ​​(​y2 ​ ​​− ​y1​ ​​)​​​ 2 ​ ​​

___________________________

B(7 ; –2) = (𝑥𝑥2 ; y2)

=√ ​​ ​​(3 − ​(​​ − 4​)​​)​​​ 2 ​+ ​​(− 4 − 1)​​​ 2​ ​​ ______________

= ​​√ (​​ 7)​​​ 2​+ ​​(− 5)​​​ 2​ ​​ _

= ​​√ 74 ​​

_______________________

AB = ​​√ ​​(​𝑥​ 2 ​​− ​𝑥​ 1 ​​)​​​ 2​+ ​​(​y2 ​ ​​− ​y1​ ​​)​​​ 2 ​ ​​

= 8,60232…

___________________________

=√ ​​ ​​(7 − ​(​​− 1​)​​)​​​ 2 ​+ ​​(− 2 − 4)​​​ 2​ ​​

= 8,6 units

______________

= ​​√ (​​ 8)​​​ 2​+ ​​(− 6)​​​ 2​ ​​ _

= ​​√ 100 ​​

= 10 units 12

D(3 ; –4) = (𝑥𝑥2 ; y2)

3.

Calculate the lengths of the line segments in the following sketch. Leave your answers in the simplest surd form. y-axis

B(–1 ; 7)

4.

E(3 ; 9)

F(–1 ; 1) C(–1 ; –1) D(–2 ; –3)

The point B(​b​; –3) is 5 units from the origin.

Find the value of ​b​.

5 units

(0 ; 0)

x-axis

Note the difference between B and​ b​.

The capital letter (B) indicates the point, and the lower case letter (b) indicates a coordinate (in this case the 𝑥-coordinate).

A(4 ; –2)

_______________________

_______________________

​​√ (𝑥2– 𝑥1)2 + (𝑦2– 𝑦1)2 ​​ =5

CD = √ ​​ ​​(​𝑥​ 2 ​​− ​𝑥​ 1 ​​)​​​ 2​+ ​​(​y2 ​ ​​− ​y1​ ​​)​​​ 2 ​ ​​

______________________

___________________________________ _______________

∴ ​​√ (𝑏– 0)2 + (–3 – 0)2 ​​ =5

_______________________

∴

)​​)​​​ 2​ + ​​(− 3 − ​ =√ ​​ ​​(− 2 − ​(​​ − 1​

(​​ − 1)​​)​​​​ 2​ ​​

∴

= ​​√ ​​(− 1 ​​​ ​+ ​​(− 2 ​​​ ​ ​​ _

)2

= ​​√ 5 ​ ​ units

B(b ; –3)

)2

∴

AB = ​​√ ​​(​𝑥​ 2 ​​− ​𝑥​ 1 ​​)​​​ 2​+ ​​(​y2 ​ ​​− ​y1​ ​​)​​​ 2 ​ ​​

∴

___________________________

=√ ​​ ​​(4 − ​(​​ − 1​)​​)​​​ 2 ​+ ​​(− 2 − 7)​​​ 2​ ​​

_

​​√ (𝑏2 + 9) ​​ =5

𝑏2 + 9 = 25 𝑏2 = 16

𝑏 = ±4

There are therefore two possible positions for B.

______________

= ​​√ (​​ 5)​​​ 2​+ ​​(− 9)​​​ 2​ ​​ _

= ​​√ 106 ​​ units

_______________________

EF = √ ​​ ​​(​𝑥​ 2 ​​− ​𝑥​ 1 ​​)​​​ 2​+ ​​(​y2 ​ ​​− ​y1​ ​​)​​​ 2 ​ ​​ ______________________

=√ ​​ ​​(− 1 − 3)​​​ 2​+ ​​(1 − 9)​​​ 2​ ​​ _______________

= ​​√ ​​(− 4)​​​ 2​+ ​​(− 8)​​​ 2​ ​​ _

© Optimi

= ​√ 80 ​​ units

13

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Facilitator's Guide 2/2 5.

_

Given: A(–7; m ​ ​), B(–3; 4) and AB = 4​​√ 5 ​​.

6.

Calculate the value of ​m​. Show your answer on a sketch.

A(–7 ; m) =(𝑥𝑥1 ; y1)

A’(–7 ; m) =(𝑥𝑥2 ; y2)

∴

∴

∴ ∴

y-axis

4 √5

B(–3 ; 4) =(𝑥𝑥2 ; y2)

4 √5

_

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

P’(p ; –2) = (x2 ; y2) = (–4 ; –2)

______________________________

)2

​∴ ∴

48 = ​​m​​ 2​​– 8m

​​m​​ 2​​– 8m – 48 = 0

P (p ; –2) = (x2 ; y2) = (2 ; –2)

________________________________

_

_________________

​√ 1+ 9 ​ = ​√ 1+ ​p​​ 2​ + 2p + 1

​​ 10 = ​​p​​ 2​​+ 2p + 2

∴

​​p​​ 2​​+ 2p – 8 = 0

∴

p = –4 or p = 2

∴ ​​(p + 4)​​​​(p – 2)​​= 0

∴ (​​ m – 12)(​​​​ m + 4)​​= 0 ∴

–2

)​​)​​​ 2​+ ​​(− 2 − ​ ​∴ ​√ ​​(0 − ​(​​ − 1​)​​)​​​ 2​+ ​​(2 − ​(​​ − 1​)​​)​​​ 2​ ​ = ​√ ​​(p − ​(​​ − 1​ (​​ − 1​)​​)​​​ 2​ ​ ​

4​​√ 5 ​​ = √ ​​ ​​(–3 – (–7))​​​ ​ + ​​(4 – m ​​​ ​ ​​ 80 = 16 + 16 – 8m + m​​ ​​ 2​​

x-as 𝑥-axis

AM = AP

_______________ 2

y-as

A(–1 ; –1) =(x1 ; y1)

_______________________

Find the value of​ p​and also show your answer on a sketch.

M(0 ; 2) =(x2 ; y2)

AB = ​​√ ​​(​𝑥​ 2 ​​− ​𝑥​ 1 ​​)​​​ 2​+ ​​(​y2 ​ ​​− ​y1​ ​​)​​​ 2 ​ ​​

A(–1;–1) is equally far from M(0; 2) and P(​p​; –2).

m = 12 or m = –4

There are therefore two possible positions for P. (See sketch.)

There are therefore two possible positions for A: A​​(–7 ; 12)​​ and A'​​(–7 ; –4)​​

14

Given

7.

A circle with the origin as its centre is given. A(3; 4) and B(-4; q ​ ​) are points on the circumference of the circle.

Calculate the value of ​q​.

8.

Given: P(2; –3), M(–2; 1) and N(𝑥; –7). N is equally far from P and M.

Draw a sketch to represent the information given. Then find the value of 𝑥 and show your answer on the sketch. y-axis

y-axis A(3 ; 4)

B(–4 ; q)

M(–2 ; 1) P(2 ; –3)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥 N(𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = (–6 ; –7)

y-axis

∴ ∴ ∴

Radiuses are equal

______________

______________

_______________

Given

∴√ ​​ ​​(–2 – 𝑥)​​​ 2​ + ​​ =√ ​​ ​​(2 – 𝑥)​​​ 2​ + ​​(–3 – ( –7)

(1 – (–7))​​​ 2​ ​​ )​​​ 2​ ​​

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

B(–4 ; q) =(–4 ; –3)

OA = 0B

NM = NP

A(3 ; 4)

B(–4 ; q) =(–4 ; 3)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

4 + 4𝑥 + ​​𝑥​​ 2​​+ 64 = 4 – 4𝑥 + ​​𝑥​​ 2​​ + 16 8𝑥 = –48 𝑥 = –6

_______________

∴√ ​​ (​​ 0 – 3)​​​ 2​ + ​​ (0 –4)​​​ 2​ ​​ =√ ​​ (​​ –4 – 0)​​​ 2​ + ​​ (q – 0)​​​ 2​ ​​ ∴ ∴ ∴ ∴

_

_____

​​ 9 + 16 ​​ = √ ​​ 16 + ​q​​ 2​ ​​ √ 25 = 16 + q​​ ​​ 2​​ ​​q​​ 2​​= 9

q=±3

There are therefore two possible positions for B. (See sketch.)

© Optimi

15

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Facilitator's Guide 2/2 9.

A(0; 0), B​​(p; q)​​and C​​(–q; p)​​are the vertices of a triangle. Prove by calculation that ∆ABC is isosceles and right-angled.

In order to prove that a triangle is isosceles, we prove that two sides are equal. _______________________

A(0 ; 0)

AC = √ ​​ (𝑥2– 𝑥1)2 + (𝑦2– 𝑦1)2 ​​ ____________________

=√ ​​ (𝑝 – 0)2 +(𝑞 – 0)2 ​​ ___________

=√ ​​ 𝑝2 + 𝑞2 ​​

_______________________

AB = √ ​​ (𝑥2– 𝑥1)2 + (𝑦2– 𝑦1)2 ​​ ______________________

B(p ; q)

=√ ​​ (–𝑞 – 0)2 +(𝑝 – 0)2 ​​ _

=√ ​​ 𝑝2 + 𝑞2 ​​

C(–q ; p)

∴ AC = AB

∆ABC is therefore right-angled and isosceles.

In order to prove that a triangle is right-angled, we prove that one angle is 90° (i.e. that two sides are perpendicular to each other). ​𝑦​ ​​ – ​𝑦​ ​​

​𝑦​ ​​ – ​𝑦​ ​​

2 1 ​​m​ AB​​​= ​​ _____ ​​ ​𝑥​ ​​ – ​𝑥​ ​​ 2

q – 0

2 1 ​​m​ AC​​​= ​​ _____ ​​ ​𝑥​ ​​ – ​𝑥​ ​​

1

2

p – 0

=_ ​​p – 0 ​​

1

=_ ​​–q – 0 ​​

q

p

=_ ​​p ​​

=_ ​​–q ​​ ​​m​ AB​​​. ​​m​ AC​​​ q

p

=_ ​​ p ​​. ​​ _ –q ​​ = –1 ∴ AB ⊥ AC

16

10.

_

_

A(0; 0), B(​​√ 3 ​; 1) and C(​​√ 3 ​; –1) are the vertices of a triangle.

Prove by calculation that ΔABC is equilateral.

√

___________________

√

_

=√ ​​ 4 ​​

= 2 units

​∴ AB = AC = BC = 2​

C(√3 ; –1)

11.

_______________ 2 _

= ​​ (​​ √ ​ 3 ​ – 0)​​​ ​ + ​​ (1 – 0)​​​ 2​ ​​

_

=√ ​​ 0 + 4 ​​

B(√3 ; 1)

_____________________ 2 _ _

= ​​ ​​(√ ​ 3 ​ – ​√ 3 ​) ​​​ ​ + ​​(–1 – 1)​​​ 2​ ​​

A(0 ; 0)

2 ​AB​= √ ​​ ​​(​𝑥​ 2​​ – ​𝑥​ 1​​)​​​ 2​ + ​​

(​𝑦​ 2​​ – ​𝑦​ 1​​)​​​ ​ ​​

___________________

2 ​BC​= √ ​​ ​​(​𝑥​ 2​​ – ​𝑥​ 1​​)​​​ 2​ + ​​

(​𝑦​ 2​​ – ​𝑦​ 1​​)​​​ ​ ​​

_

=√ ​​ 3 + 1 ​​

​ΔABC ​is therefore equilateral.

A parallelogram with vertices A(–2; 2), B(3; 2), C(1; –2) and D(–4; p) is given.

Calculate the value of ​p​.

_

A(–2 ; 2)

=√ ​​ 4 ​​

= 2 units

___________________

D(–4 ; p)

_______________ 2 _

(–1 – 0)​​​ 2​ ​​ = ​​ (​​ √ ​ 3 ​ – 0)​​​ ​ + ​​

_

=√ ​​ 3 + 1 ​​

C(1 ; –2)

REMEMBER

_

=√ ​​ 4 ​​

The opposite sides of a parallelogram are equal and parallel.

= 2 units © Optimi

B(3 ; 2) 𝑥𝑥𝑥𝑥𝑥𝑥𝑥

2 ​AC​= √ ​​ ​​(​𝑥​ 2​​ – ​𝑥​ 1​​)​​​ 2​ + ​​

(​𝑦​ 2​​ – ​𝑦​ 1​​)​​​ ​ ​​

√

y-axis

17

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Facilitator's Guide 2/2 ​mAB ​ ​​ = ​mDC ​ ​​​ ​ ∴

Opposite sides of parallelogram are parallel

− 2 − p

2 − 2 ​ _

​ = ​_

​ 3 − ​(​​ − 2​)​​ 1 − ​(​​ − 4​)​​

​∴ 5​(− 2 − p)​ = 0 ​

12.

​∴

12.1

__________________

=√ ​​ ​​(–2 – ( –5)) ​​​ 2​ + ​​(1 – 3)​​​ 2​ ​​

_

=√ ​​ 9 + 4 ​​

Cross multiply

_

=√ ​​ 13 ​​ units

​∴ − 10 − 5p = 0​ ​∴

___________

AM = ​​√ (𝑥2 – 𝑥1)2 + (y 2 – y1)2 ​​

___________

BM = ​​√ (𝑥2 – 𝑥1)2 + (y 2 – y1)2 ​​

− 5p = 10​

p = − 2​

_

_

=√ ​​ 13 ​​ units

Prove that the circle passes through the points A(–5; 3) and B(1; 3).

_

​∴​AM = BM = ​​√ 13 ​​ units

y-axis

12.2

B(1 ; 3) M(–2 ; 1)

_

=√ ​​ 9 + 4 ​​

A circle with M(–2; 1) as its centre and radius ​√ 13 ​ units are given.

A(–5 ; 3)

__________________

=√ ​​ ​​(–2 – 1)​​​ 2​ + ​​ (1 – 3)​​​ 2​ ​​

The circle therefore passes through points A and B.

Calculate the circumference and area of this circle if the units are in centimetres. Round off your answers to the nearest integer.

Circumference of circle = 2𝜋​​r​​ ​​ _

= 2𝜋 ×​ √ ​ 13 ​​

= 23 cm ​​to the nearest integer Area of circle = 𝜋​​r​​ 2​​ _

= 𝜋​​​(√ ​ 13 ​) ​​​ ​​

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

2

= 40,84070...

= 41 ​​cm​​ 2​​to the nearest integer

_

For the circle with centre M and radius √ ​​ 13 ​​ ​​units to pass through _ points A and B, we must prove that AM = BM = ​​√ 13 ​​ ​​units, so that radiuses are equal.

18

9.2 THE MID-POINT OF A LINE SEGMENT

Worked example 3

In the diagram below, C(𝑥; y) is the mid-point of line segment AB, with A(​𝑥1​; y​1​) and B(​𝑥2​; y2​​).

y₁

Solution

(​𝑥​ ​​ + ​𝑥​ ​​ ​𝑦​ ​​ + ​𝑦​ ​​ ) 2 + (–8) ______ 5 + (–1) = ​​(______ ​ 2 ​ ; ​ 2 ​ )​​

B(𝑥𝑥₂ ; y₂)

𝑥𝑥₂

(

𝑥𝑥₁

M(–3 ; 2)

A(2 ; 5) 𝑥𝑥𝑥𝑥𝑥𝑥𝑥

B(–8 ; –1)

= ​​(–3 ; 2)​​

Worked example 4

The 𝑥-coordinate of C is exactly halfway between ​𝑥​1 and ​𝑥​2 on the ​𝑥​-axis. 𝑥 + 𝑥

If ​M​(− 2; 1)​ is the mid-point of the line segment between ​A​(​​−6; 8​)​​​​ and ​B​(​​a; b​)​​​​, find the coordinates of B ​ ​.

The value of the ​𝑥​-coordinate of C is therefore ​​_ 1 2 2​​ ​​. This is the average value of ​𝑥​1 and ​𝑥​2. y1 + y2 In the same way, the value of ​y​is ​​ _

​​ ​​.

2

A(–6 ; 8)

It follows that:

y-axis

M(–2 ; 1)

The formula for calculating the mid-point, C, of a line segment between two points, A(𝑥1; y1) and B(𝑥2; y2), is: 𝑥 + 𝑥 y1 + y2 CAB = ​​(_ ​ 1 2 2​ ; ​_ 2 ​ )​​

B(a ; b)

The word “mid-point” indicates that a line segment is bisected. © Optimi

)

4 = ​​ ______ ​ –6 ​ ; ​ ______ ​ ​​ 2 2

C(𝑥𝑥𝑥𝑥𝑥y) 𝑥𝑥

y-axis

1 ​​M​ AB​​​= ​​ ______ ​ 1 2 2 ​ ; ​ ______ 2 ​ ​​ 2

A(𝑥𝑥₁ ; y₁)

y y₂

Find the mid-point of the line segment between A(2;5) and B(–8;–1) and show your answer on a sketch.

19

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Facilitator's Guide 2/2

( ​𝑥​ ​​ + ​𝑥​ ​​

y + y

2 _ 2 M ​​(–2 ; 1)​​= ​​[_ ​ 1 2 ​

; ​ 1 2 ​

]​​​​

𝑥 + 𝑥

a – 6 ∴ –2 = ​ ​_ ​​ 2

AND

∴ – 4 = a – 6 ∴ a = 2 ∴ B​​(2 ; –6)​​

D(d ; –4)

(​𝑥​ ​​ + ​𝑥​ ​​

​𝑦​ ​​ + ​𝑦​ ​​

)

(​𝑥​ ​​ + ​𝑥​ ​​

​𝑦​ ​​ + ​𝑦​ ​​

)

(​𝑥​ ​​ + ​𝑥​ ​​

​𝑦​ ​​ + ​𝑦​ ​​

)

(

= (​​ 4 ; 3)​​

1=_ ​​ b + 8 ​​ 2

∴ 2 = 𝑏 + 8

∴ 𝑏 = – 6

( 1 = (​​ 1 ; –3​ _ ​ ​​ 2)

)

( = (​​ –4 ; ​_ 12 ​)​​

)

–4 – 3 = ​​ ______ ​ –3 + 5

​ ; ​ ______

​ ​​ 2 2

1 ​​M​ CD​​​= ​​ ______ ​ 1 2 2 ​ ; ​ ______ 2 ​ ​​ 2

–4 + 5 = ​​ ______ ​ –4 – 4

​ ; ​ ______

​ ​​ 2 2

y-axis A(–3 ; 9)

1 ​​M​ EH​​​= ​​ ______ ​ 1 2 2 ​ ; ​ ______ 2 ​ ​​ 2

(

F(5 ; m) E(g ; k)

)

1 ​​M​ EF​​​ = ​​ ______ ​ 1 2 2 ​ ; ​ ______ 2 ​ ​​ 2

Consider the sketch below. Find the coordinates of the mid-point of each of the following line segments: AB, CD, EF and EH. C(–4 ; 5)

)

9 – 3 = ​​ ______ ​ –3 + 11

​ ; ​ ______ ​ ​​ 2 2

Exercise 9.2: The mid-point of a line segment 1.

​𝑦​ ​​ + ​𝑦​ ​​

1 1 ​M​ AB​​​= ​​ ​ ______ 2 ​ ; ​ ______ 2 ​ ​​ 2 2

Solution

)

–4 – 4 = ​​ ______ ​ –3 + 11

​ ; ​ ______

​ ​​ 2 2

= (​​ 4 ; –4)​​

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

B(11 ; –3) H(a ; b)

20