Mathematics Grade 10 Study Guide 2/2

Study Guide 2/2

10

Mathematics

CAPS

Mathematics FET Phase Grade 10 • Study Guide 2/2 home classroom college workplace

2210-E-MAM-SG02

9 781990

948510

Mathematics Study Guide 2/2 – Grade 10

2210-E-MAM-SG02

9 781990

948510

CAPS-aligned

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

10.1 Interest...........................................................................................................27

TABLE OF CONTENTS

Exercise 10.1: Quiz.................................................................................................... 28

10.2 Simple interest.............................................................................................28 PREFACE..................................................................................................................... 1

Exercise 10.2................................................................................................................ 31

10.3 Hire purchase...............................................................................................31

THEME 9: ANALYTICAL GEOMETRY......................................................................... 6

Exercise 10.3................................................................................................................ 34

Introduction....................................................................................................................6

Self-evaluation............................................................................................................ 36

Revision exercise...........................................................................................................8

9.1

The distance formula................................................................................... 8

9.2

Exercise 9.1: The distance formula.................................................................... 10

10.4 Compound interest.....................................................................................36

Exercise 10.4................................................................................................................ 39

10.5 Timelines.......................................................................................................39

The mid-point of a line segment............................................................11

Exercise 10.5................................................................................................................ 42

Exercise 9.2: The mid-point of a line segment.............................................. 13

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

Self-evaluation............................................................................................................ 15

9.3

The gradient (inclination) of a line.......................................................15

9.4

Exercise 9.3: The gradient (inclination) of a line......................................... 17

Points on a straight line (collinear points).........................................19

9.5

Exercise 9.4: Points on a straight line (collinear points).......................... 20

10.6 Comparing financial situations..............................................................44

Exercise 10.6................................................................................................................ 46

10.7 Inflation and population growth............................................................47

Exercise 10.7................................................................................................................ 51

10.8 Foreign exchange rates..............................................................................52

The equation of a straight line ...............................................................21

Exercise 10.8................................................................................................................ 54

Exercise 9.5: The equation of a straight line.................................................. 21

Self-evaluation............................................................................................................ 55

Self-evaluation............................................................................................................ 22

Summary of theme.................................................................................................... 55

Summary of theme.................................................................................................... 23

End of theme exercise.............................................................................................. 56

End of theme exercise.............................................................................................. 23

THEME 11: STATISTICS...........................................................................................59

THEME 10: FINANCE AND GROWTH.....................................................................25

Introduction................................................................................................................. 59

Introduction................................................................................................................. 25

Revision exercise........................................................................................................ 60

Revision exercise........................................................................................................ 25 © Optimi

i

G10 – Mathematics

G10 – Mathematics – Study Guide 2/2 11.1 Grouped data................................................................................................64

THEME 12: TRIGONOMETRY: PROBLEMS IN TWO DIMENSIONS..................... 97

Exercise 11.1: Grouped data................................................................................. 67

Introduction................................................................................................................. 97

11.2 Measures of central tendency in grouped data.................................70

Revision exercise........................................................................................................ 97

11.2.1 Estimated mean of grouped data...........................................................70

12.1 Problems involving angles of elevation and depression................99

11.2.2 Estimated median of grouped data.......................................................70

Exercise 12.1: Problems involving angles of elevation and depression......................................................................................................... 102

11.2.3 Modal class of grouped data....................................................................70

12.2 Problems involving bearings (direction)......................................... 104

Exercise 11.2: Measures of central tendency in grouped data............... 73

Exercise 12.2: Problems involving bearings (direction)........................ 106

Self-evaluation............................................................................................................ 74

Self-evaluation......................................................................................................... 110

11.3 Measures of spread.....................................................................................75 11.3.2 Interquartile range (IQR).........................................................................76

Summary of theme................................................................................................. 110

11.3.3 Semi-interquartile range..........................................................................76

THEME 13: EUCLIDEAN GEOMETRY....................................................................113

11.3.4 Percentiles.....................................................................................................76

Introduction.............................................................................................................. 113

11.3.1 Quartiles.........................................................................................................75

End of theme exercise........................................................................................... 110 Exercise 13.1............................................................................................................. 113

11.3.5 Outliers...........................................................................................................77

Self-evaluation......................................................................................................... 118

Exercise 11.3: Measures of spread..................................................................... 83

Summary of theme................................................................................................. 118

11.4 The five-number summary......................................................................84

Exercise 11.4: The five-number summary...................................................... 85

THEME 14: MEASUREMENT.................................................................................119

Exercise 11.5: Analysing statistical summaries and graphs.................... 90

14.1 Revision: Perimeter, area, surface area and volume.................... 123

End of theme exercise.............................................................................................. 92

14.1.3 Nets of 3D shapes.....................................................................................123

Introduction.............................................................................................................. 119

Self-evaluation............................................................................................................ 87

11.5 Analysing statistical summaries and graphs .....................................87

Revision exercise..................................................................................................... 121

14.1.1 Perimeter/circumference and area................................................... 123

Self-evaluation............................................................................................................ 91

14.1.2 Regular polygons......................................................................................123

Summary of theme.................................................................................................... 92

14.1.4 Surface area and volume........................................................................124

ii

Exercise 14.1: Revision: Perimeter, area, surface area and volume.. 124

14.2 Surface area and volume of cones.......................................................127

15.2.2 Definition of relative frequency...........................................................147

Exercise 14.2: Surface area and volume of cones..................................... 129

Exercise 15.2: Relative frequency.................................................................... 147

14.3 Surface area and volume of pyramids...............................................129

Self-evaluation......................................................................................................... 148

Exercise 14.3: Surface area and volume of pyramids........................... 131

15.3 Venn diagrams...........................................................................................148

Self-evaluation......................................................................................................... 135

15.3.3 Using Venn diagrams to determine probability............................. 152

Exercise 14.5: Effect on volume and surface area when any dimension is multiplied by a constant factor k......................137

15.4.1 The union of two events.........................................................................155

14.4 Surface area and volume of spheres.................................................. 133

15.3.1 Using Venn diagrams to represent events........................................ 148

Exercise 14.4: Surface area and volume of spheres................................. 134

15.3.2 Union and intersection...........................................................................149

multiplied by a constant factor ​k​................................................................ 135

15.4 Probability identities..............................................................................155

Self-evaluation......................................................................................................... 137

15.4.3 Complementary events...........................................................................158

14.5 Effect on volume and surface area when any dimension is

Exercise 15.3: Venn diagrams............................................................................ 154

15.4.2 Mutually exclusive events......................................................................157

Summary of theme................................................................................................. 138

15.4.4 Mutually exhaustive events...................................................................159

End of theme exercise........................................................................................... 138

Exercise 15.4: Probability identities, mutually exclusive events and complementary events................................................................................ 160

THEME 15: PROBABILITY.....................................................................................140

Self-evaluation......................................................................................................... 161

Introduction.............................................................................................................. 140

15.1 Theoretical probability..........................................................................143

Summary of theme................................................................................................. 161

15.1.1 Terminology...............................................................................................143

PAPERS...................................................................................................................165

15.1.2 Expressing probability...........................................................................144

FORMULAE..............................................................................................................176

Revision exercise..................................................................................................... 141

End of theme exercise........................................................................................... 162

15.1.3 Definition of probability........................................................................144

EUCLIDIAN GEOMETRY: ACCEPTED REASONS..................................................179

Exercise 15.1: Theoretical probability........................................................... 146

INDEX......................................................................................................................182

15.2 Relative frequency...................................................................................146 15.2.1 The difference between relative frequency and theoretical probability...........................................................................146

© Optimi

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 – Study 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:

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.

1.

revision of the concepts, definitions and procedures required as previous knowledge, 2. an exercise or activity with solutions so that you can test your previous knowledge yourself, or 3. a combination of the above.

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.

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.

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.

© Optimi

3

G10 – Mathematics

G10 – Mathematics – Study 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.

© Optimi

5

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Study Guide 2/2

THEME 9

What is analytical geometry about?

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

ANALYTICAL GEOMETRY

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.

Introduction

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

In this theme you 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.

Background

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

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.

Prior knowledge

In order to master this theme, you 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: y – y 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 1

• know the properties of triangles and quadrilaterals.

6

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

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

–10

III

y 10

II

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

5

–5

P(3 ; 5)

I x

5

–5 –10

(0 ; 0) origin

10

IV 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

​

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

​y​ ​​ − ​y​ ​​

​= ​_ ​𝑥2​ ​​ − ​𝑥1​ ​​ ​

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

​

2

1 − 3 ​= ​_ − 8 − 0

​

​y​ ​​ − ​y​ ​​

​= ​_ ​𝑥2​ ​​ − ​𝑥1​ ​​ ​ 2

​= ​_ − 84 ​

1

​= − _ ​12 ​

​= ​_ 5 − 0 ​ 3 − 0 ​= ​_ 53 ​

© Optimi

1

7

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Study Guide 2/2

Solutions

Revision exercise

1.

A(–1; 3), B(2; –1), C(8; 1), D(5; 5)

Gradient of BC = _ ​​ 8 – 2

​​= _ ​​ 26 ​​= _ ​​ 31 ​​

2.

Consider the figure and answer the questions that follow.

5 – 3 Gradient of AD = _ ​​5 – ( – 1)

​​= _ ​​ 26 ​​= _ ​​ 31 ​​​​​​​​ 1 – ( – 1)

The gradients (inclinations) are equal.

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

3.

4.

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

​​ = _ ​​ –4 ​​​​ ​​ 3

The gradients (inclinations) are equal.

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

9.1 THE DISTANCE FORMULA 1.

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

4.

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

2.

3.

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

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

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

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.

REMEMBER

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

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.

8

Worked example 1

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

(Pythagoras)

= 4² + 5²

= 16 + 25

Find the distance between A and B.

= 41

Solution

_

​∴​ distance = ​√ 41 ​

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

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

AB

REMEMBER

_____________ _

=√ ​​ 100 ​​

___________________

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

(​𝑦​ 2​​ – ​𝑦​ 1​​)​​​ ​ ​​

© Optimi

_________________________

= ​​√ (3–(–5))2 + (4–(–2))2 ​​ =√ ​​ (8)2 + (6)2 ​​

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

__________________________

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

= 10 units

9

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Study Guide 2/2

Important

Worked example 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.

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

Exercise 9.1: The distance formula

Solution ________________________

AB = ​​√ (𝑥2– 𝑥1 ) + (𝑦2– 𝑦1 ) ​​ 2

___________________

2

1.

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

2.

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

5=√ ​​ 𝑥²– 4𝑥 + 13 ​​

3.

25 = 𝑥²– 4𝑥 + 13

Draw the points A(–1 ; 4) and B(7; –2) on a system of axes and calculate the distance from A to B. Draw the points C(–4 ; 1) and D(3; –4) on a system of axes and calculate the distance from C to D. Calculate the lengths of the line segments in the following sketch. Leave your answers in the simplest surd form. y-axis

0 = 𝑥2 – 4𝑥 – 12

B(–1 ; 7)

0 = (𝑥 – 6)(𝑥 + 2) 𝑥 = 6 or 𝑥 = –2

F(–1 ; 1)

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

C(–1 ; –1)

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

D(–2 ; –3)

4.

E(3 ; 9)

x-axis

A(4 ; –2)

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

Find the value of ​b​.

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).

10

5.

6.

7.

_

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

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

12.

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

12.2

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

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​.

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

A(3 ; 4)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

8.

9.

10.

11.

y₁

y₂

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.

B(𝑥𝑥₂ ; y₂)

𝑥𝑥₂

_

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

Prove by calculation that ΔABC is equilateral.

C(𝑥𝑥𝑥𝑥𝑥y) 𝑥𝑥₁

𝑥𝑥

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

The value of the ​𝑥​-coordinate of C is therefore ​​_ 1 2 2​​ ​​. This is the average value of ​𝑥​1 and ​𝑥​2.

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

y1 + y2 In the same way, the value of ​y​is ​​ _

​​ .​​ 2

Calculate the value of ​p​.

© Optimi

A(𝑥𝑥₁ ; y₁)

y

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. _

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

9.2 THE MID-POINT OF A LINE SEGMENT

y-axis

B(–4 ; q)

_

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

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

11

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Study Guide 2/2 It follows that:

A(–6 ; 8)

The formula for calculating the mid-point, C, of a line segment between two points, A(𝑥1; y1) and B(𝑥2; y2), is:

M(–2 ; 1)

𝑥 + 𝑥 y1 + y2 CAB = ​​(_ ​ 1 2 2​ ; ​_ 2 ​ )​​

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

B(a ; b)

Worked example 3

Solution

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

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

𝑥 + 𝑥

(

)

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

= ​​(–3 ; 2)​​

a – 6 ∴ –2 = ​ ​_ ​​ 2

y-axis M(–3 ; 2) B(–8 ; –1)

y + y

2 2 M ​​(–2 ; 1)​​= ​​[_ ​ 1 2 ​

; _ ​ 1 2 ​

]​​​​

Solution

​𝑥​ ​​ + ​𝑥​ ​​ ______ ​𝑦​ ​​ + ​𝑦​ ​​ ​​M​ AB​​​= ​​ ______ ​ 1 2 2 ​ ; ​ 1 2 2 ​ ​​

y-axis

AND

∴ – 4 = a – 6

A(2 ; 5)

∴ a = 2 ∴ B​​(2 ; –6)​​

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

Worked example 4 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 ​ ​.

12

1=_ ​​ b + 8 ​​ 2

∴ 2 = 𝑏 + 8

∴ 𝑏 = – 6

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

2.3

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

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)

D(d ; –4)

2.

y-axis A(–3 ; 9)

F(5 ; m)

y-axis

E

3.

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

B(11 ; –3)

4.

B(–5 ; –2)

D(4 ; 6)

Given: M is the mid-point of AB and N is the mid-point of BC. Find the coordinates of N.

y-axis

M(–6 ; 3)

C(5 ; 8)

N(r ; s)

B(p ; q)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

Prove that the mid-point of BC = Q(0; 0).

Now find the coordinates of E, the mid-point of AD.

© Optimi

If M(–3; –2) is the mid-point of A(–8; –5) and B(​p​; q ​ ​), calculate the values of p ​ ​and ​q​.

A(–13 ; 5)

C(5 ; 2)

Q

“If the mid-points of a pair of opposite sides of a rectangle are connected, two rectangles are formed which are congruent.”

Tip In order to prove that two rectangles are congruent, you need to prove that the corresponding sides are equal and that the corresponding angles are equal.

H(a ; b)

E(g ; k)

Is the following statement true?

Give reasons for your answer.

Given: Rectangle ABCD with A(–6; 2), B(–5; –2), C(5; 2) and D(4; 6). In addition, AE = ED.

A(–6 ; 2)

2.1 2.2

2.4

Prove that AB = EQ = DC.

13

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Study Guide 2/2 5.

Square PQSR is given.

6.1

y-axis

P(–1 ; 4)

S(c ; d)

5.1

What are the values of ​a​, ​b​, ​c​and ​d​in the coordinates of R and S? Indicate the coordinates of R and S on the sketch.

6.

Consider triangle ABC below.

7.

Now prove that the diagonals of a square bisect each other using the mid-point formula. y-axis

B(–7 ; –2)

Indicate the median point of the triangle on the sketch.

Tip A median connects a vertex with the mid-point of the side opposite that vertex. The point where the three medians intersect is called the median point (or centroid) of the triangle.

Example:

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

R(a ; b)

5.2

6.2

Q(4 ; 4)

Redraw the sketch, do the necessary calculations and draw all three medians of the given triangle on the sketch.

A(3 ; 8)

Rhombus ABCD is given. y-axis

A(1 ; 8)

B

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

7.1 7.2

C(9 ; –6)

14

D

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

C(3 ; –8)

Prove that the point of intersection of the diagonals lies on the 𝑥-axis.

Find the length of AC rounded off to two decimal places.

9.3 THE GRADIENT (INCLINATION) OF A LINE

Self-evaluation Use the following scale to determine how comfortable you are with each topic in the table that follows:

The gradient or inclination of a line refers to the steepness of the line.

Earlier on, we defined this as:

1. Help! I am not at all comfortable with the topic; I need help.

2. Alarm! Alarm! I am not comfortable with the topic; however, I merely need more time to go through the topic again.

3. OK! I am fairly comfortable with the topic, but do get stuck occasionally.

4. Sharp! I am comfortable with the topic.

5. Whoohoo, it's party time! I am completely comfortable with the topic, and can even answer more difficult questions on this.

Complete the table.

Topic

1

2

3

4

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

​

The gradient can also be defined as:

the vertical difference between 2 points on a line _______________________________________________ ​the horizontal difference between the same 2 points on the line

​

REMEMBER

5

I understand how the distance formula is derived.

The formula for calculating the gradient of a line connecting the two points A(𝑥1; y1) and B(𝑥2; y2) is:

I can use the distance formula to determine the distance between two given points (i.e. the length of a line segment) in the Cartesian plane.

​y​ ​​− ​y​ ​​

2 1 ​m = ​ _ ​𝑥​ ​​− ​𝑥​ ​​ ​​​​ 2

y-axis

I can use the distance formula to solve problems in analytical geometry.

B(𝑥𝑥2 ; y2)

I understand how the formula for the midpoint of a line segment is derived.

(y2 – y1)

I can use the formula for the midpoint of a line segment to determine the midpoint of a line segment in the Cartesian plane.

I can use the formula for the midpoint of a line segment to solve problems in analytical geometry.

© Optimi

1

A(𝑥𝑥1 ; y1)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

(𝑥𝑥2 – 𝑥𝑥1)

y2 – y1 means “the change in y”.

We can also write this as ∆y (delta y)

15

C(𝑥𝑥2 ; y1)

𝑥2 – 𝑥1 means “the change in 𝑥”.

We can also write this as ∆𝑥 (delta 𝑥) G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Study Guide 2/2 Consider the sketch below:

Note y1 – y2 y2 – y1 _ Correct: m = _ ​​ 𝑥​​ 2 – 𝑥 1 ​​ or m = 𝑥​​ 1 – 𝑥2

y2 – y1

y1 – y2

Incorrect: m = ​ _ ​​ or m = _ ​ 𝑥 – 𝑥 ​ ​ 𝑥 – 𝑥 1

2

2

y-axis

𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥

The order in which the 𝑥- and y-coordinates are subtracted from each other is important:

y=3

1

3

Worked example 5 Find the inclination and mid-point of the line segment connecting ​A​(​​4; 1​)​​​​ and​​ B​(​​−8; 4​)​​​​.

Inclination

Mid-point

( (

) )

​𝑥​ ​​ + ​𝑥​ ​​ ______ ​𝑦​ ​​ + ​𝑦​ ​​ M(𝑥 ; 𝑦) = ​​ ______ ​ 1 2 2 ​ ; ​ 1 2 2 ​ ​​ 4 + (–8) _ = ​​ _ ​ 2 ​ ; ​ 1 + 4 ​ ​​ 2 1 = ​​ –2 ; 2​ _ ​ ​​ 2

​𝑦​ ​​ – ​𝑦​ ​​

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

–4

1

4 – 1 = ​​_ –8 – 4

​​

(

3 = ​​_ –12 ​​

= ​​_ –1 ​​ 4

)

Everywhere on this vertical line, the 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

In the sketch we see:

• The y-coordinates of points that lie alongside each other on a horizontal line are equal.

• The 𝑥-coordinates of points that lie directly below each other on a vertical line are equal.

Horizontal and vertical lines

REMEMBER

A line which is parallel to the 𝑥-axis is called a horizontal line.

The gradient of a horizontal line is zero, because there is no vertical change in the gradient of the line:

The gradients of parallel lines are equal: ​m1 ​ ​​ = ​m2​ ​​​

The gradients of perpendicular lines: ​m1 ​ ​​ × ​m2​ ​​ = − 1​

change in y

0 ​m = ​_________________ change in 𝑥

​ = ​ _________________

​ = 0​ change in 𝑥

A line which is parallel to the y-axis is called a vertical line.

The gradient of a vertical line is undefined, because there is no horizontal change in the gradient of the line: change in y

Everywhere on this horizontal line, the y-value is 3.

change in y

​m = ​_________________ change in 𝑥

​ = ​ _________________

​ = undefined​

0

16

3.

Exercise 9.3: The gradient (inclination) of a line 1.

Consider the line segments given below:

In the sketch, AB ⊥ CD and PQ ⊥ RS. The points A(–14; 5), B(2; 1), R(5; –2) and S(–4; –6) are given. Find the gradients of CD and PQ. y-axis

A(–14 ; 5)

C(–8;p)

1.1 1.2 2.

C

D(q;–4)

4.

Find all the unknown coordinates and indicate these on the sketch.

Now find the gradient of each line segment.

Consider the sketch given below: A(a ; 5)

y-axis

5.

B(b ; 3) –2

C(m ; n) D(d ; –4)

–1

P(𝑥𝑥 ; y)

E(1 ; e)

Q S(–4 ; –6)

17

P

Calculate the value of ​a​.

If AB​ ⊥ ​CD with A(–5; 2), B(​b​; –1), C(–4; –3) and D(–1; 3), calculate the value of​ b​.

A(–5 ; 2)

Q(r ; s)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥 R(5 ; –2)

The points ​A​(​​–5 ; –1​)​​​​and ​B​(​​–3 ; a​)​​​​form line segment ​AB​and the points ​​C​(​​–4 ; –3​)​​ ​​and D ​​ ​(​​–1 ; 3​)​​​​ form line segment C ​ D​.

C(–4 ; –3)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥 F(k ; h)

B(2 ; 1)

Line segment A ​ B​and ​CD​are given, with ​AB ∥ CD​.

5

Find the values of the coordinates indicated by the letters a ​ ​to ​s​.

© Optimi

D

B(b ; –1)

D(–1 ; 3)

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Study Guide 2/2 6.

Quadrilateral ABCD is given.

8.

y-axis B(5 ; 4)

A(–5; 4)

C(3 ; –3)

D(–7 ; –3)

6.1 6.2

B(–10 ; 3)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

Prove that AD​ ∥ ​BC.

9.

Prove that AD = BC.

A(1 ; 6)

C(1 ; –3)

D(–2 ; 3)

C(–6 ; –5)

ABCDEF is a regular hexagon. Do the necessary calculations to prove that all three pairs of opposite sides are parallel. y-axis

B(–6 ; 1) C(–3 ; –3)

D(4 ; 4) B(–2 ; –1)

y-axis

A(–3 ; 5)

Consider quadrilateral ABCD. Show by calculation that the quadrilateral is a trapezium. y-axis

A(–6 ; 5)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

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

6.4 Determine in which quadrant the point of intersection of the diagonals lies. Show your calculations.

7.

Consider the sketch and use analytical methods to prove that the diagonals of kite ABCD intersect perpendicularly.

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

18

F(2 ; 5) E(5 ; 1) D(2 ; –3)

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

10.

In the triangle below, N is the mid-point of AB and P is the mid-point of AC. Use analytical methods to prove that ​NP ∥ BC​and ​NP = ​_12​ BC​.

A(–2 ; 4)

9.4 POINTS ON A STRAIGHT LINE (COLLINEAR POINTS)

y-axis

Points are collinear if they lie on the same straight line.

We can prove this using one of the following methods: 1.

2.

𝑥𝑥𝑥𝑥𝑥𝑥𝑥

B(–6 ; –4)

C(6 ; –2)

Worked example 6

Prove that A(2; –1), B(5; 2) and C(7; 4) are collinear.

REMEMBER

Solution

According to the mid-point theorem (which was dealt with in Theme 8), the following applies: If a line segment is drawn from the mid-point of one side of a triangle to the midpoint of another side of the triangle, this line is parallel to the third side and equal to half of the third side.

If D is the mid-point of AB, for e𝑥ample, and E is the mid-point of AC, it follows that DE​ ∥ ​BC and DE = ​​12_ ​​BC.

11.

by proving that the gradients between any two pairs of the three points are equal by proving that the distance between the two furthest points is equal to the sum of the distance between the three points.

A

D

A

The three points will lie on the same straight line if mAB = mBC.

E

B

C

B

We are therefore going to prove that ​mAB ​ = ​​ ​mBC ​ ​​​(gradient method): 2 –(–1)

mAB = ​​ _

​​ 5 – 2

C

= ​​_ 33 ​​ =1

Sketch the points A(–5; 6), B(–5; –8) and C(6; –8). Determine which angle in triangle ABC is 90° using analytical methods.

4 –2 but mBC = ​​ _ ​​ 7 – 5

=_ ​​ 22 ​​ =1

∴ mAB = mBC

∴ A, B and C are collinear.

Alternative follows on next page

© Optimi

19

G10 – Mathematics

Theme 9: Analytical geometry

G10 – Mathematics – Study Guide 2/2 OR:

Worked example 7

We can also prove that AB + BC = AC (distance method): _______________

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

The points A(-6; 2), B(2; -2) and C(-3; p) are collinear. Find the value of ​p​.

_

=√ ​​ 9 + 9 ​​

Solution

_

=√ ​​ 18 ​​

If A, B and C lie on the same straight line, the gradient of AB is the same as the gradient of BC.

_

= 3​​√ 2 ​​

_______________

∴

BC = √ ​​ ​​(7 – 5)​​​ 2​ + ​​ (4 – 2)​​​ 2​ ​​

𝑝– (–2) –2 – 2 ∴ ​​ ________

​​ = ​​________ –3 – 2 ​​ 2 – (–6)

_

=√ ​​ 4 + 4 ​​ _

= 2​​√ 2 ​​

_______________

_

=√ ​​ 25 + 25 ​​ _

=√ ​​ 50 ​​ _

_

_

𝑝 + 2

∴

–4 ______ ​​ ___ ​​ = ​​ –5 ​​ 8

∴

1 𝑝 = ​​ __ ​​ 2

∴ 8𝑝 + 16 = 20

AC = √ ​​ ​​(7 – 2)​​​ 2​ + ​​( 4 –(–1))​​​ 2​ ​​ = 5​​√ 2 ​​

𝑚AB = 𝑚BC

_

​∴ ​AB + BC = ​3 ​√ 2 ​ + 2 ​√ 2 ​ = 5 ​√ 2 ​ = AC Therefore A, B and C are collinear.

1.

Determine whether the following points lie on the same straight line: A(–1; 2), B(2; 5) and C(0; 3). Use the gradient method.

3.

Determine whether the following points lie on the same straight line: A(–1; –1), B(2; 8), C(3; 11) and D(1; 5).

2.

20

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

Determine whether the following points lie on the same straight line: A(1; 5), B(2; 7) and C(3; 6). Use the distance method.