Collins IGCSE Maths Revision Guide Sample

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Cambridge IGCSE™

Cambridge IGCSE™ Maths

Clear and concise syllabus coverage, with the Extended material clearly differentiated

•

Topics in short, user-friendly sections to help you plan your revision in manageable chunks

•

Worked examples and guidance on structuring your answers

•

Support for key subject vocabulary and exam command words

•

Quick tests and exam-style practice questions for every topic

•

A complete set of exam-style practice papers to help you prepare for assessment

•

A free ebook version, so you can revise from anywhere

Maths REVISION GUIDE

REVISION GUIDE

•

Cambridge IGCSE™ Maths

Collins Cambridge IGCSE™ Maths Revision Guide provides clear and accessible revision content to support all students. With focused revision for every topic, plus four separate practice opportunities, use this revision guide to build your confidence and help you prepare for your Cambridge IGCSETM Maths assessments.

Made with responsibly sourced paper and vegetable-based ink.

This text has not been through the Cambridge International endorsement process.

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Chris Pearce, Andrew Milne

20/5/2024 11:06 pm

CONTENTS Revise

p. 6

p. 32

Sets

p. 8

p. 32

Powers and roots

p. 10

p. 33

Fractions, decimals and percentages

p. 12

p. 34

Ordering and the four operations

p. 14

p. 34

Indices and standard form

p. 16

p. 35

p. 18

p. 36

p. 20

p. 36

p. 22

p. 37

p. 24

p. 38

p. 26

p. 39

p. 28

p. 40

p. 30

p. 41

Introduction to algebra

p. 42

p. 68

Algebraic manipulation

p. 44

p. 69

Algebraic fractions

p. 46

p. 70

Further indices

p. 48

p. 70

Equations

p. 50

p. 71

Inequalities

p. 52

p. 72

Sequences

p. 54

p. 72

Proportion

p. 56

p. 74

Graphs in practical situations

p. 58

p. 75

Graphs of functions

p. 60

p. 76

Sketching graphs

p. 62

p. 77

ES

Types of number

G

Practise

SA

Section 1: Number

Estimation and limits of accuracy Ratio and proportion

Percentages Using a calculator Exponential growth and decay

E

Surds

PA

Rates

M

PL

Section 2: Algebra and graphs

3

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Differentiation

p. 64

p. 78

Functions

p. 66

p. 79

Linear graphs 1

p. 80

p. 84

Linear graphs 2

p. 82

p. 85

Section 3: Coordinate geometry

ES

Section 4: Geometry Geometric terms and constructions Scale drawings

p. 98

p. 88

p. 99

p. 90

p. 100

p. 92

p. 102

G

Similarity

p. 86

PA

Symmetry Angles

p. 94

p. 103

p. 96

p. 104

p. 106

p. 116

Perimeter and area

p. 108

p. 117

Circles, arcs and sectors

p. 110

p. 118

Surface area and volume

p. 112

p. 119

Compound shapes and solids

p. 114

p. 120

Right-angled triangles

p. 122

p. 130

Trigonometric functions

p. 124

p. 131

Non-right angled triangles

p. 126

p. 131

Trigonometry in three dimension

p. 128

p. 132

Circle theorems

E

Section 5: Mensuration

SA

M

PL

Units of measure

Section 6: Trigonometry

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CONTENTS Section 7: Transformations and vectors Transformations

p. 134

p. 138

Vectors

p. 136

p. 139

Introduction to probability

p. 140

p. 144

Probability of combined events

p. 142

p. 145

Section 8: Probability

Statistical data Averages Charts and diagrams

p. 160

p. 150

p. 161

p. 152

p. 162

p. 154

p. 163

p. 156

p. 164

p. 158

p. 165

PA

Scatter diagrams

p. 148

G

ES

Section 9: Statistics

Cumulative frequency

p. 167

Mixed exam-style questions: Extended

p. 179

Practice Paper 1: Non-calculator (Core)

p. 187

Practice Paper 2: Non-calculator (Extended)

p. 195

Practice Paper 3: Calculator (Core)

p. 204

Practice Paper 4: Calculator (Extended)

p. 213

List of formulas (Core)

p. 224

List of formulas (Extended)

p. 225

Answers

p. 226

Glossary/Index

p. 238

M

PL

E

Mixed exam-style questions: Core

SA

Histograms

5

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Section 2: Algebra and graphs

E 1

Proportion The mass of a coin is directly proportional to its diameter.

A coin with a diameter of 24 mm has a mass of 7.2 g.

[3 ]

Find the mass of a coin with a diameter of 20 mm. _

​ y​is directly proportional to √ ​ 2x + 5 ​.​ When ​x = 2,​ ​y = 60.

2 Find ​y​when ​x = 10.​

[3 ]

ES

Show me _

Write the inequality as an equation: ​y = k​√

_

Substitute values of x and y: ​60 = k​√............

G

So k = ...... and the equation is y = ............

PA

When x = 10, y = ............ ​ y ∝ ​x​3​.​Find the values of ​a​and ​b​in this table.

3 4

5

b​ ​

​y​

960

a​ ​

7680

E

x​ ​

[5 ]

​ y​is inversely proportional to ​x​2​.​ When ​x = 0 . 2,​ ​y = 2 . 5.​ 4 Find ​y​when ​x = 0 . 4.​

b

Find ​x​when ​y = 0 . 4.​

PL

a

[2 ]

[3 ]

​ y​is inversely proportional to the square root of ​x.​ 6

SA

NC

M

1 ​.​ When ​x = 1,​ ​y = 10.​ y ∝ _ ​ ​ 5 5x − 1 Find ​y​when ​x = 5.​

[3 ]

When ​x = 2,​ ​y = 12.​ Find the exact value of ​y​when ​x = 16.​

[3 ]

74

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PRACTISE

Graphs in practical situations 1

This graph can be used to convert from km/h to m/s.

m/s

20 15 10 5 0

0

10

20

30

40

50

60

70

80

ES

km/h

Use the graph to convert 54 km/h to m/s.

b

Use the graph to convert 10 m/s to km/h.

c

Convert 60 m/s to km/h.

2

This graph shows the journey of a cyclist from Acton to Corlon via Byfield.

[1 ] [1 ] [2 ]

PA

50 Corlon 40 30 Byfield 20

E

Distance (km)

G

a

Acton 0 08 00

PL

10 09 00

10 00

11 00 Time

12 00

13 00

Write down the time the journey started.

[1 ]

b

Find the average speed of the journey from Acton to Byfield.

[2 ]

c

How long did the cyclist stop at Byfield?

[1 ]

d

Work out the average speed from Byfield to Corlon.

[3 ]

SA

M

a

Work out the average speed for the whole journey.

3

This graph shows the speed of a car over a 15-second interval.

Speed (m/s)

e

24 16

0

a

[3 ]

0

5 15 Time (s)

Work out the acceleration for the first 5 seconds.

[2 ]

Acceleration is the gradient of the speed-time graph

75

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Section 2: Algebra and graphs

b

Work out the acceleration for the last 10 seconds.

[2 ]

c

Calculate the distance travelled.

[4 ]

Graphs of functions 1

Copy and complete this table of values of ​y = 8 − x​ ​2​

a

​x​

–4

​y​

–3

–2

–1

1

–1

2

3

4

7

Use your table to draw a graph of ​y = 8 − ​x​2​.​

b

[3 ]

c

On the same axes draw the line ​y = x.​

[2 ]

d

Use the graph to solve the equation ​8 − ​x​2​ = x.​

ES

Use a scale of 1 cm to 1 unit on the ​x​-axis and 1 cm to 2 units on the ​y​-axis.

a

​x​

1

2

3

4

​y​

5

6

7

G

10 Copy and complete this table of values of ​y = _ ​ x .​​

8

9

2.5

1.67

3

​x​

–2

–1

0

1

2

3

​y​

4

[2 ] [3 ]

–6

Use your table to draw a graph of ​y = ​x​3​ − ​x​2​ − 6x.​

b

M

Use a scale of 2 cm to 1 unit on the ​x​-axis and 1 cm to 5 units on the ​y​-axis. c

On the same axes draw the line ​y = 3x + 6.​

d

Use your graph to solve the equation ​x​3​ = ​x​2​ + 9x + 6.​

SA

E

[3 ]

4

PL

–3

E

c

[2 ]

1.43

10 Draw a graph of ​y = ​_ x ​for ​1 ≤ x ≤ 10​. Use a scale of 1 cm to 1 unit on each axis.

5 On the same axes draw a graph of ​y = _ ​x ​.​ Copy and complete this table of values of ​y = ​x​3​ − ​x​2​ − 6x.​ a

b

[2 ]

10

PA

2

E

0

[2 ]

4

[3 ] [3 ] _

Copy and complete this table of values of ​y = x + 2 √ ​ x ​.​

a

​x​

0

​y​

1

2

3

4

4.83

[3 ]

[3 ]

5 9.47 _

Use your table to draw a graph of ​y = x + 2 √ ​ x ​.​

b

Use a scale of 2 cm to 1 unit on the ​x​-axis and 1 cm to 1 unit on the ​y​-axis. _

Use the graph to solve the equation √ ​ x ​ = 3 . 5 − 0 . 5x.​

c

[3 ] [3 ]

Show me _

_

​ x ​ = x + 2​(. . . . . . . . . . . .)​ = . . . . . . .​ If √ ​ x ​ = 3 . 5 − 0 . 5x​then ​y = x + 2 √ From the graph, if y = …. then x = ….

76

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PRACTISE

Sketching graphs Sketch the graph of ​y = 10 − 2x.​

1

a

b

Find the equation of this graph.

[2 ] [2 ]

y

3

ES

a

b

Write ​x​2​ + 2x − 15​in completed square form.

c

Sketch the graph of ​y = ​x​2​ + 2x − 15.​ y

3

A

B

[2 ] [3 ]

C

x

E

0

[3 ]

G

E

Solve the equation ​x​2​ + 2x − 15 = 0.​

2

PA

E

x

0

−3

D

PL

The equation of this curve is ​y = ​x​2​ − 8x + 7.​ Find the coordinates of A, B, C and D.

E

a

b

Sketch the graph of ​y = ​x​3​ − 4x.​

5

4 ​x ​. Find the values of ​a​and ​b.​ a ​ (​ 10, a)​and (​ 20, b)​are on the graph of ​y = 3 + _ 4 Find the asymptotes of ​y = 3 + _ ​x ​.​ 4 Sketch the graph of ​y = 3 + _ ​x ​.​

SA

b c

E

Factorise ​x​3​ − 4x​as much as possible.

4

M

E

6

Sketch the graph of ​y = 2 ​ ​x​for ​− 2 ≤ x ≤ 2.​

[6 ] [2 ] [3 ] [2 ] [2 ] [3 ] [3 ]

77

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Section 2: Algebra and graphs

E 1

Differentiation a

6 Draw the curve ​y = _ ​x ​for ​1 ≤ x ≤ 6.​ [3 ]

b

Draw the tangent to the curve at (​ 3, 2)​.​

[2 ]

c

Use the tangent to estimate the gradient of the curve at (​ 3, 2)​.​

[2 ]

2

[2 ]

b

The equation of a curve is ​y = ​x​3​ − 2 ​x​2​ + 4x.​ dy Find ​_​.​ dx Find the gradient of the curve where it crosses the ​y​-axis.

3

The equation of a curve is ​y = 5x − ​x​2​.​

a

ES

Use a scale of 1 cm to 1 unit on each axis.

Show me

The point required is (… , …) dy _ ​ ​ = . . . . . . . . .​ dx

The gradient at (… , …) is …..

PL

The equation of the tangent is …….

E

Factorise, ​x​(. . . . . . . .)​ = 0​ so x = …. or ….

PA

Where the curve crosses the x-axis ​5x − x​ ​2​ = 0​

[5 ]

G

Find the equation of the tangent to the curve at the point where it crosses the positive x​ ​-axis.

[2 ]

[2 ]

[2 ]

b

The equation of a curve is ​y = x​ ​3​ − 3 ​x​2​ + 3.​ dy Find ​_​.​ dx Find the gradient at ​(1, 2)​.​

c

Find the coordinates of the stationary points.

[4 ]

d

State whether each stationary point is a maximum of a minimum.

a b

SA

5

M

The equation of a curve is ​y = ​x​3​ − 6 ​x​2​ + 10x.​ dy Find ​_​.​ dx Find the coordinates of the two points on the graph where the gradient is 1.

4

a

Give reasons for your answers.

[5 ]

[1 ]

[3 ]

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PRACTISE E

Functions

12 ​ f(​ x)​ = _ ​ x ​ + 2​ 1 a

Find ​f​(4)​.​

[1 ]

b

Find ​ff​(3)​.​

[2 ]

c

Find ​f​−1​(x)​.​

[2 ]

d

When the domain of ​f​(x)​is {​ x : x ≥ 1}​find the range of ​f(​ x)​.​

[2 ]

f​(x)​ = 10 − 2x​and ​g(​ x)​ = (​ x − 3)​2​.​ ​ 2 b

Find ​fg​(6)​.​

c

Find ​f​−1​(x)​.​

d

Solve the equation ​g​(x)​ = 4.​

f(​ x)​ = 3x + 4​and ​g(​ x)​ = 5x − 2.​ ​ 3 Find ​gg​(2)​.​

b

Find ​fg​(x)​.​Write your answer as simply as possible.

c

Find ​fg​(x)​ + gf​(x)​.​

d

Solve the equation ​f​−1​(x)​ = 3.​

E

10 ​ ​ ​ h(​ x)​ = _ 4 x−3 Find ​h​(− 7)​.​ a

PA

a

Find ​h​−1​(x)​.​

c

Solve the equation ​h​(x)​ = x.​

PL

b

G

Find ​f​(3)​.​

ES

[1 ]

a

[2 ] [2 ] [2 ]

[2 ] [2 ] [2 ] [2 ]

[1 ] [2 ] [4 ]

​ f(​ x)​ = ​x​2​ − 5​and ​g(​ x)​ = 2x + 3.​ 5

[2 ]

Find ​gf​(4)​.​

b

Solve the equation ​f​(x)​ = 4.​

[2 ]

c

Solve the equation ​f​(x)​ = g​(x)​.​

[4 ]

SA

M

a

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Section 3: Coordinate geometry

Linear graphs 1

Syllabus links:

C3.1; E3.1; C3.2; E3.2; C3.3; E3.3; C3.5; E3.5; C3.6; E3.6

Learning aims:

• Draw straight-line graphs • Find the gradient and the equation of a straight-line graph • Find equations of parallel lines The equation of a straight line ​y = 2x − 1​is the equation of a straight line. x​ ​ ​y​

–2 –5

–1 –3

0 –1

1 1

3 5

4 7

5 9

4 The gradient of the line is _ ​ ​ = 2.​ 2 The ​y​-intercept where the line crosses the ​y​-axis is –1.

Here is a graph:

G

y 6

Both these numbers are in the equation ​ y = 2x − 1​

PA

5 4 4

3

The gradient is 2.

2 1

2

y​ = mx + c​where ​m​ and ​c​are numbers is the equation of a straight line. The gradient is ​m​ and the ​y​-intercept is ​c.​

3

x

4

E

1

Key Point

The ​y​-intercept is –1.

2

−3 −2 −1 0 −1

PL

−2

Example 1

Find the gradient of this straight line.

M

a)

2 3

ES

Here is a table of values:

y

4

SA

3 2 1

−2 −1 0 −1

1

2

3

4

5

6

7

8

x

−2

b)

Write down the y-intercept.

c)

Find the equation of the line.

−3 1 gradient = _ ​ ​ = − ​_​​ Use the triangle formed by the axes.

6 2 b) The ​y​-intercept is 3.

Where the line crosses the ​y​-axis.

1 y = −_ ​ ​x + 3​ ​y = (​ gradient)​x + (​ y-intercept)​ c) ​ 2

a)

80

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REVISE Parallel lines Parallel lines have the same gradient. y

y=

4 3 2

y=

1 −3 −2 −1 0 −1

2 x+2 3 2 y= x 3

1

2

3

4

2 x−3 3 x

5

−2

ES

−3

2 2 2 y = _x − 3 y = _x y = _x + 2 3 3 3 2 All these lines have a gradient of _ . They are parallel.

3

PA

G

Example 2 Find the equation of the line parallel to y = 3x + 5 that passes through (4, 8).

The coefficient of x.

The equation is y = 3x + c.

The equation always has this form.

8 = 3×4+c

Substitute the coordinates (4, 8).

c = 8 − 12 = −4

Solve the equation to find c.

PL

The equation is y = 3x − 4.

E

The gradient is 3.

Quick Test

x

0

2

4

–2

4

10

SA

y

M

1. Find the gradient of the line y = 6x − 8.

2. a) Use this table of coordinates to draw a straight line.

b) Find the gradient of the line. c) Find the equation of the line parallel to y = −x that passes through (4, 2).

81

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Section 3: Coordinate geometry

E Linear graphs 2

Syllabus links: E3.1; E3.2; E3.3; E3.4; E3.5; E3.6; E3.7

Learning aims:

• • •

E Straight-line graphs and their equations in various forms E Gradient, length and midpoint of a line segment E Parallel and perpendicular lines

Equation of a straight line graph

The equation can be written in different ways such as: 2 ​y = 4 − _ ​ ​x​ or ​3y = − 2x + 12​ or ​2x + 3y = 12​ 3 Example 1 The equation of a line is ​4x + 5y = 20.​

ES

2 2 ​y = − _ ​ ​x + 4​is the equation of a straight line with a gradient of ​− _ ​ ​​and 3 3 a ​y​-intercept of 4.

Draw a graph of the line.

b)

Find the gradient of the line.

a)

If ​x = 0​then ​5y = 20​and ​y = 4​ The easiest way is to find where the line crosses each axis.

PA

G

a)

y

2 0

2

x

6

PL

4

E

4

​ (​ 0, 4)​is on the line.

If ​y = 0​then ​4x = 20​and ​x = 5​

M

​ (​ 5, 0)​is on the line.

SA

b) ​ 4x + 5y = 20​ You can find the gradient from the graph but you can also find it by rearranging the equation to make ​y​ the subject. ​ 5y = − 4x + 20​ 4 ​ y = −_ ​ ​x + 20​ 5

Line segments The gradient of the line through A and B is: y A (3, 6)

6

a 0 −2

x 9 B (9, −2)

3 b

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REVISE difference between y-coordinates a −2 − 6 −8 4 _ ​ ​ = ____________________________ ​ ​ = _ ​ ​ = ​_​ = − ​_​ b

difference between x-coordinates

9−3

6

3

Key Point

Using Pythagoras’s theorem, the length of the line segment _

_

​ ​8​2​ + ​6​2​ = 10​ is ​√​a​2​ + ​b​2​ = √

If ​A​(​x​1​, ​y​1​)​and ​B(​ ​x​2​, ​y​2​)​​ are two points:

3+9 _ 6 + −2 The midpoint of the line segment = ​(_ ​ ​, ​ ​ ​ = (​ 6, 2)​ 2 2 )

Parallel and perpendicular lines A line parallel to L has the same gradient, ​m.​ 1 A line perpendicular to L has the gradient ​− _ ​m ​.​ A line L has the equation ​2x + 5y = 40.​

PA

Example 2

G

The line L has the equation ​y = mx + c.​The gradient of the line is ​m.​

ES

​y​ ​ − y​ ​ ​

2 1 ​​x​ ​ − x​ ​ ​​ • The gradient of the line through A and B is _ 2 1 __________________ 2 ​x​2​ − ​x​1​)​ ​ + (​ ​y​2​ − y​ ​1​)​2​ • The length of the line segment AB is √​ ​( ​x​1​ + x​ ​2​ ​y​1​ + ​y​2​ ​ ​, _ ​ ​​ • The midpoint of AB is (​ _ 2 2 )

Find the equation of the line perpendicular to L passing through ​(3, 4)​.​

Rewrite the equation to make ​y​the subject Here are the two lines in the worked example.

E

2 ​5y = 40 − 2x​ ​⇒ y = 8 − _ ​ ​x​ 5

M

4 ​ = 2 . 5 × 3 + c​

PL

2 Gradient of L = ​− _ ​ ​​ 5 Gradient of perpendicular line 5 =_ ​ ​ = 2 . 5​ The coefficient of ​x.​ 2 ​y = 2 . 5x + c​ The equation of the perpendicular line Substitute the coordinates and find ​c.​

The perpendicular line is ​y = 2 . 5x − 3 . 5​

This could be written as ​2y = 5x − 7​

SA

​c = 4 − 7 . 5 = − 3 . 5​

y

8 (3, 4) 0

20

x

−3.5

Quick Test

1. Find the gradient of the line through (​ − 6, 0)​and (​ 0, 9)​.​ 2. Find the gradient of the line with equation ​6x + 5y = 12.​ 3. Find the equation of a line perpendicular to ​y = 10 − 2x​that crosses the ​x​-axis at (​ 6, 0)​.​ 4. P is (​ − 4, 2)​and Q is (​ 1, 14)​.​ a) Find the length of PQ. b) Find the midpoint of PQ.

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Section 3: Coordinate geometry

Linear graphs 1 y

1 3 2 1

−2 −1 0

1

2

3

4

5

x

a

Find the gradient of this line.

[2 ]

b

Find the equation of the line.

[2 ]

ES

Show me Two points on the line are (0, 1) and . . . . .

Find the equation of the parallel line through (​ 0, 4)​.​

2 1

3

7

​y​

5

3

–1

Use this table of coordinates to draw a straight line.

[3 ]

b

Find the gradient of the line.

[2 ]

c

Find the ​y​-intercept.

d

Write down the equation of the line.

[1 ]

3

The equation of a straight line is ​y = 4x + 10.​

a

Find the gradient of the line.

b

Find the coordinates of the point where it crosses the ​y​-axis.

[1 ]

c

Find the equation of a parallel line through the origin (​ 0, 0)​.​

[1 ]

E

a

PL

​x​

[2 ]

PA

c

G

The gradient is ______ ​​ . . - 1 ​​ = . . . ... -0

[1 ]

SA

M

[1 ]

d

Find the equation of a parallel line through (​ 2, 0)​.​

4

a

b

Find the equation of a parallel line with a ​y​-intercept of –4.

[1 ]

c

Find the equation of a parallel line through (​ 1, 2)​.​

[2 ]

5

a

b

Find the gradient.

6

c

When the line is extended it passes through (10, a). Find the value of a.

5

[2 ]

Find the equation of a straight line with a gradient of –2 and a ​y​-intercept of 8.

Write down the y-intercept of this line.

[2 ]

[1 ]

y

[2 ]

4

[2 ]

3 2 1 −2 −1 0

1

2

3

4

5

x

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PRACTISE E

Linear graphs 2

1

A is (​ − 10, 2)​and B is (​ − 1, 14)​.​

a

Find the gradient of AB.

[2 ]

b

Find the length of AB.

[2 ]

Use Pythagoras’ theorem to find the length.

Find the midpoint of AB.

2

y

[2 ]

ES

c

3 2 1 1

2

3

x

G

−3 −2 −1 0 −1

[2 ]

Find the gradient of this line.

b

Find the equation of this line.

c

Find the equation of a perpendicular line through (​ 2, 3)​.​

3

The equation of a line is ​2x + 3y = 15.​

a

The point ​(a, 8)​is on the line. Find the value of ​a.​

b

Find the equation of a parallel line through (​ 6, 4)​.​

[3 ]

c

Find the equation of a perpendicular line through (​ 6, 2)​.​

[3 ]

4

The line ​3x + 4y = 18​crosses the ​x​-axis at A and the ​y​-axis at B.

a

Find the coordinates of A and B.

[2 ]

b

Find the length of AB.

[2 ]

c

Find the gradient of AB.

[2 ]

d

Find the equation of the line perpendicular to AB and passing through B.

[3 ]

SA

M

PL

E

PA

a

[2 ] [3 ]

[2 ]

5

A line segment joins A (3, -5) and B (6, 7)

a

Find the midpoint of the line segment AB

[2 ]

b

Find the equation of the straight line through A and B.

[3 ]

6

The coordinates of P are (2, 5)

The coordinates of Q are (8, 1) Work out the equation of the perpendicular bisector of PQ.

[4 ]

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Section 4: Geometry

Similarity

Syllabus links: C4.4; E4.4

Learning aims:

• Calculate length in similar shapes • E Understand the relationship between lengths and areas in similar shapes • E Understand the relationship between lengths and volumes in similar solids Similar shapes

C2 53°

These two shapes are similar.

15 cm

C1

B ​ ​ ​​C​ ​

2 2 21 ​ ​ = 1 . 5​ ​_​ = _

​B​1​​C​1​

127°

21 cm

12 cm

A1

8 cm

B1

PA

15 2 2 _ ​= _ ​ ​ = 1 . 5​ ​

127°

14 cm

8 cm

14

​ ​​ ​ ​ ​D C

D2

G

The ratio between corresponding sides is always the same: ​A​2​​B​2​ 12 _ ​ ​ = 1 . 5​ ​= _ ​ 8 ​A​1​​B​1​

D1

53°

ES

10 cm

Corresponding angles are the same.

A2

12 cm

B2

​ ​1​D C ​ ​1​ 10 ​ ​2​is an enlargement of ​A​1​​B​1​​C​1​​D​1​with a scale factor of 1.5 A ​ ​2​​B​2​​C​2​D

a)

E

Example 1 Explain why these triangles are similar.

A

30 cm

42°

C

85° 25 cm 53° X

SA

Find ​x.​

42°

x cm

85° y cm

b)

Z

M

B

PL

53° 20 cm

Y

30 cm

c)

Find ​y.​

a) The angles of the triangles are the same. This is sufficient to show that to triangles are similar.

25 XY _ b) ​ ​ ​= _ ​ ​ = 1 . 25​ ​XY​and ​AB​are corresponding 20 AB sides because they are both opposite 42°. x XZ _ ​ ​ = 1 . 25​ ​XZ​and ​AC​are corresponding ​ ​ ​= _ 30 AC sides. ​ x = 1 . 25 × 30 = 37 . 5​ Solve the equation for ​x.​ 30 YZ _ ​ y ​ = 1 . 25​ ​YZ​and ​BC​are corresponding c) ​ ​ ​= _ BC sides.

30 ​ y = _ ​ ​ = 24​ Solve the equation for ​y.​ 1 . 25

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REVISE E Similar areas C2

If two shapes are similar with a length scale factor of ​k​then the area scale factor is ​k​2​.​

53°

The area of ​A​1​​B​1​​C​1​​D​1​in the first diagram is ​ 88 ​cm​​2​.​

15 cm

C1 10 cm

The linear scale factor is 1.25 so the area scale factor is ​1. ​25​​2​.​

D1

The area of ​A​2​​B​2​​C​2​​D​2​is ​88 × 1. ​25​​2​ = 198 ​cm​​2​.​

53°

D2

127°

127° 21 cm

14 cm

8 cm 12 cm

The area of triangle ABC in the first worked example is ​240 ​cm​​2​.​

A1

8 cm

B1

ES

The linear scale factor is 1.25 The area of DEF is ​240 × 1. ​25​​2​ = 375 ​cm​​2​.​

A2

E Similar volumes

12 cm

B2

A

PA

B

Example 2

G

Two solids are similar when the ratio of corresponding sides are the same. One is an enlargement of the other.

Height (cm)

Area of base (​​cm​​2​​)

PL

Shape

E

These two solids are similar. Complete this table.

A

20

The linear scale factor ​= 24 ÷ 20 = 1 . 2​

If the ratio of corresponding sides in two similar solids is ​k​then the ratio of corresponding areas is ​k​2​and the ratio of the volumes is ​k​3​.​

300

24

M

B

Volume (​​cm​​3​​)

Key Point

8640

The heights are lengths.

SA

Area scale factor ​= 1. ​2​2​ = 1 . 44​

Area of base of B ​= 300 × 1 . 44 = 432 c​ m​​2​​ Multiply by the area factor.

Volume scale factor ​= 1. ​2​3​ = 1 . 728​ Volume of A ​= 8640 ÷ 1 . 728 = 5000 ​cm​​3​​

Divide the find the volume of A.

Quick Test 1. The shortest side of a triangle is 24 mm and the longest side is 38 mm.

The shortest side of a similar triangle is 84 mm. Find the longest side.

E 2. A trapezium 32 cm wide has an area of ​1440 ​cm​​2​.​

Find the area of a similar trapezium 18 cm wide.

E 3. A pyramid 15 cm high has a volume of ​800 c​ m​​3​.​

Find the volume of a similar pyramid with a height of 45 cm.

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Section 4: Geometry

Symmetry

Syllabus links: C4.5; E4.5

Learning aims:

• Recognise lines of symmetry and rotational symmetry in two dimensions • E Recognise planes of symmetry and rotational symmetry in three dimensions Symmetry of shapes Shapes can have reflection symmetry or rotation symmetry.

Here are the symmetries of some common shapes.

Isosceles triangle

Equilateral triangle

1 line of symmetry

3 lines of symmetry

no rotational symmetry

rotational symmetry of order 3

Square

Rhombus

Parallelogram

2 lines of symmetry

no lines of symmetry

rotational symmetry of order 2

rotational symmetry of order 2

Regular pentagon

Circle

ES

G

PL

Kite

PA

rotational symmetry of order 4

E

4 lines of symmetry

Rectangle

1 line of symmetry

rotational symmetry of order 5

rotational symmetry of order 2

every diameter is a line of symmetry

SA

M

no rotational symmetry

5 lines of symmetry

2 lines of symmetry

E Symmetry of solids

Solids can have:

• reflection symmetry with a plane of symmetry • rotation symmetry with an axis of symmetry.

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Section 4: Geometry

Circle theorems

Syllabus links: C4.7; E4.7; E4.8

Learning aims:

• Use geometrical properties of circles to find angles • E Use symmetry properties of circles Semicircles and tangents

S

​O​is the centre of the circle.

C

​AB​is a diameter.

A

​ST​is the tangent at ​A.​

ES

• The angle in a semicircle at ​C​is 90°.

• The angle between the tangent ​AT​and the diameter ​AB​is 90°.

G

E Geometrical properties ​ ​is the centre of the O circle. ​ OQ​is the angle at the P centre.

x°

​ RQ​is the angle at the P circumference.

y° B

The angle at the centre is twice the angle at the circumference:

The angles at ​A,​ ​B​ and ​C​are all equal.

Q

Angles at the circumference in the same segment are equal.

P

PL

Q

C

A

E

O

​y = 2x​

d°

b° B

SA

D

​ BCD​is a cyclic A quadrilateral.

M

A a°

The vertices are all on a circle.

c°

The sum of opposite angles of a cyclic quadrilateral is 180°:

C

a + c = 180 and b + d = 180

Q R

T

​ ,​ ​B​and ​C​ are A angles at the circumference from P​and ​Q.​

B

PA

R

O

B

​C​is an angle in a semicircle.

S​ T​is a tangent to the circle at ​P.​

a°

a° P

T

Angles ​TPQ​and ​ PRQ​are angles in the same segment.

Angles in the same segment are equal.

S

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