01/11/20151 All of the ship would remain visible as apparent size diminishes. Whereas on a flat Earth

20/04/23 1

All of the ship would remain visible as apparent size diminishes.

Whereas on a flat Earth

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10 cm

15 cm

20 cm

30 cm

10

15

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The lantern throws a shadow across the floor.

What would happen if the lantern was closer to the man?

What would happen if it was further away?

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(4, 6)(6, 5)(3, 2)(1, 5)(-4, 3)

(-1, 1)(5, 3)(7, 1)(-7, 2)(-2, -1)

(-4, -2)(4, -2)(2, -3)(-2, -4)

y

x 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

–7 –6 –5 –4 –3 –2 –1 -1-2-3-4-5-6

Starter Activity 2 What are the coordinates?

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Learning Objectives

• To enlarge the given image

• To find the centre of the enlargement

Keywords: Enlargement, scale factor (SF), image, object, centre of enlargement (COE)

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• Scale Factor (SF): the size by which an image is enlarged.

• SF can be a positive or negative, whole number or fractional number.

• Centre of Enlargement (COE): COE is the point from which the image is enlarged.

• If the COE is not given, we can draw the image anywhere.

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20/04/23 1020/04/23 Similar Shapes 10

I want you to enlarge the rectangle.

How many times bigger do you want it ?

Twice as big. That’s a Scale Factor = 2

Do you mean twice the AREA or twice the line lengths ?

You know exactly what I mean !

O.K. You mean twice the line lengths !

20/04/23 1120/04/23 Similar Shapes 11

What are the DIMENSIONS of the following enlargements ?

S.F. = 2

S.F. = 2

S.F. = 3

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Copy the following shapes onto squared paper and sketch their enlargements.

S.F. = 3

S.F.= 4

S.F.= 2

S.F. = 3

S.F.= 3

1) 2) 3)

4)

5)

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If I wanted to enlarge a shape by say a Scale Factor = 2 I could draw it anywhere !

I will fix the position of the ‘image’ by using a Centre of Enlargement in the same way that the position of a lens fixes the

position of an image.

xC of E4

Scale Factor = 2

x 2 = 8x

8 7

x 2 = 14x

4.47x 2 = 8.94

x

7.28x 2 = 14.56

x

All of the ‘light rays’ originate from and lead back to the Centre of Enlargement.

If the enlargement is carried out on a grid then you may prefer to ‘count the squares.’

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xC of E

Scale Factor = 2

xx

xx

If the enlargement is carried out on a grid then you may prefer to ‘count the squares.’

4 leftx 2 = 8 left

4 left and 2 downx 2 = 8 left and 4 down

7 leftx 2 = 14 left


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To enlarge the rectangle by scale factor x2 from

the point shown.

Centre of Enlargement

Object

A B

CD

Or Count Squares

Image

A/ B/

C/D/

Enlargements from a Given Point

1. Draw the ray lines through vertices.

2. Mark off x2 distances along each line.

3. Draw and label image.

X2

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D

To enlarge the kite by scale factor x3 from the

point shown.


ObjectA

B

C

Or Count Squares

Image

A/

B/

C/

D/



2. Mark off x3 distances along lines from C of E.


X3

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To enlarge the triangle by scale factor x4 from the

point shown.


Or Count SquaresObject

A

B

C

Image

C/

A/

B/

X4

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Negative Enlargements from a Given Point

1. Draw ray lines from each vertex through C of E.

2. Mark off x1distances along lines from C of E.

3. Draw and label image.B/

A/

C/

Image

Image is Inverted

D/

D


AB

CObject

To enlarge the kite by scale factor -1 from the

point shown.

Or Count Squares

-1

20/04/23 21

D


AB

CObject


point shown.

Or Count Squares


2. Mark off distances x2 along lines from C of E.


B/

A/

D/

C/

Image

Image is Inverted


-2

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Fractional Enlargements from a Given Point

1. Draw ray lines from C of E to each vertex.

2. Mark halfway distances along lines.


B

A

D

C

Object

Or Count Squares

To enlarge the kite by scale factor x½ from the

point shown.


A/

B/

D/

C/

Image

½

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2. Mark off 1/3 distances along lines.


A’ B’

C’D’

Image

BA

D C

Or Count Squares

To enlarge the rectangle by scale factor x1/3 from

the point shown.


Object

1/3

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the point shown.


Object

A B

CD

Image

A/ B/

C/D/



2. Mark off x 2 distances along each line.


No Grid 1

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D


point shown.


ObjectA

B

C

Image

A/

B/

C/

D/





No Grid 2

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point shown.


ObjectA

B

C

Image

C/

A/

B/

No Grid 3

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2. Mark off x1distances along lines from C of E.

3. Draw and label image.B/

A/

C/

Image

Image is Inverted

D/

D


AB

CObject


point shown.

No Grid 4

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D


AB

CObject


point shown.


2. Mark off distances x2 along lines from C of E.


B/

A/

D/

C/

Image

Image is Inverted


No Grid 5

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Enlargement

20/04/23 30x2 across

1 up

Enlarge by a scale factor of 2 about the centre of enlargement x.

4 across

2 up3

26

4

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Enlarge by a scale factor of ½ about the centre of enlargement x.

x

6 down

7 across

3 down

3.5 across

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x x

Scale Factor = 3 and centre of origin = x

2.2

2.2 x 3 = 6.6x

1.4

1.4 x 3 = 4.2

x

2.2

2.2 x 3 = 6.6

x

or

1 right and 2 upx 3 = 3 right and 6 up

x

1 right and 1 downx 3 = 3 right and 3 down

x2 right and 1 down

x 3 = 6 right and 3 down

x

ImageObject

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Scale Factor = 2 centre of enlargement = x

x x3.6

3.6 x 2 = 7.2

x

3.2

3.2 x 2 = 6.4x

2.2

2.2 x 2 = 4.4

x

1.4

1.4 x 2 = 2.8x

3 left and 2 upx 2 = 6 left and 4 up

x


x

1 right and 2 upx 2 = 2 right and 4 up

x

1 right and 1 downx 2 = 2 right and 2 down

x

20/04/23 3420/04/23 Similar Shapes 34

Enlarge the following shapes by the given Scale Factors.

x S.F. = 2

1)

XS.F. = 3

2)

x

S.F. = 2

3) XS.F. = 3

4)

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Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit)

xS.F. = 2

( - 9 , 7 ) y

x

1)

Object

Image

Object

Image

2)

S.F. = 3

x

( 3 , 4 )

3)

Object

Image

S.F. = 4x

( - 4 , - 2 )

20/04/23 36

To enlarge a shape on a centimetre grid, simply multiply the lengths by the scale factor.

Hint: You only need to worry about the vertical and horizontal lengths, the diagonals will follow.

2cm

3cm

6cm

9cm

Scale Factor = 3

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To enlarge a shape about a centre of enlargement, draw lines from the centre through the vertices.

Scale Factor = 3

Now measure along the lines three times the original distance from the centre of enlargement to each vertex. This is where the corresponding vertex will appear. Tip: You could use compasses.


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aa’

Scale Factor = 2

The original vertices should be labeled with normal letters. The corresponding vertices on the image should be labeled with dashed letters.

20/04/23 39


What if the CoE is inside the shape?

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What about if I need to find the centre of enlargement?

0 1 2 3 4 5 6 7 8 9 x

1

98765432

y

We have found the centre of enlargement!

(2, 1)

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Scale Factor = -1

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Enlarge this triangle by a scale factor of 3.

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Enlarge this triangle by a scale factor of 3.

12

X 3

X 33

6

20/04/23 44

Enlarge this triangle by a scale factor of 3 using (2, 1) as the centre of enlargement.

2

1

20/04/23 45


2

1

20/04/23 46


2

1

1. Draw lines from the centre of enlargement through the vertices (corners) of the shape.

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2

1

Draw lines from the centre of enlargement through the vertices (corners) of the shape.

2. Use the lines to find the corners of the enlarged shape

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2

1

Draw lines from the centre of enlargement through the vertices (corners) of the shape.

Use the lines to find the corners of the enlarged shape

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Scale Factor = -1

20/04/23 51

What about if I need to find the centre of enlargement?

0 1 2 3 4 5 6 7 8 9 x

1

98765432

y

We have found the centre of enlargement!

(2, 1)

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20/04/23 5320/04/23 Similar Shapes 53

Scale Factor = 1/2

x

10.2½ of 10.2 = 5.1

x

16.1½ of 16.1 = 8.05

x

10.2½ of 10.2 = 5.1

x

16.1½ of 16.1 = 8.05

x

Scale factors less than 1 will produce images smaller than their objects.

C of EObject Image

You may prefer to count squares !

xObject

10 left and 2 up½ of 10 left and ½ of 2 up = 5 left and 1 up

x

10 left and 2 down½ of 10 left and ½ of 2 down = 5 left and 1 down

x

16 left and 2 up½ of 16 left and ½ of 2 up = 8 left and 1 up

x

16 left and 2 down½ of 16 left and ½ of 2 down = 8 left and 1 down

xImage

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x

S.F.= 1/2

1)

x

2)

S.F.= 1/4

3)

S.F.= 2/3

x

S.F.= 1 5. x

4)

20/04/23 5520/04/23 Similar Shapes 55


Object

Image

1)

S.F. = 1/2

C of E at ( - 6 , 0 )x

x

Object

Image

S.F. = 1/3C of E at ( 3 , 2 )

2)

Object

ImageS.F. = 1/4xC of E at ( - 1 , - 1 )3)

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20/04/23 5720/04/23 Similar Shapes 57

The lens in your eye produces an image using a Negative Scale Factor !

x

Scale Factor = - 2

2.83 x -2 = - 5.66

x2.83 x -2 = - 5.66

x

4.47x -2 = - 8.94

x

Object

Image

As with the other enlargements you could have carried out these negative enlargements by counting squares in the opposite directions.

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x

S.F. = - 2

1)

XS.F. = - 0.5

2)

20/04/23 5920/04/23 Similar Shapes 59


x

Object

Image

1) Enlargement, Scale Factor = - 3

Centre of Enlargement at ( - 7 , 4 )

x

Object

Image

Enlargement, Scale Factor = - 1/4

2) Centre of Enlargement at ( 5 , - 1 )

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2. Mark halfway distances along lines.


B

A

D

C

Object


point shown.


A/

B/

D/

C/

Image

No Grid 6

20/04/23 61



2. Mark off 1/3 distances along lines.


A’ B’

C’D’

Image

BA

D C

To enlarge the rectangle by scale factor x 1/3 from the point shown.


Object

No Grid 7

20/04/23 62


the point shown.


Object

A B

CD

Or Count Squares


Worksheet 1

20/04/23 63

D


point shown.


ObjectA

B

C

Or Count Squares


Worksheet 2

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point shown.


Or Count SquaresObject

A

B

C

Worksheet 3

20/04/23 65


D


AB

CObject


point shown.

Or Count Squares

Worksheet 4

20/04/23 66

D


AB

CObject


point shown.

Or Count Squares


Worksheet 5

20/04/23 67


B

A

D

C

Object

Or Count Squares


point shown.


Worksheet 6

20/04/23 68


BA

D C

Or Count Squares

To enlarge the rectangle by scale factor x 1/3 from the point shown.


Object

Worksheet 7

20/04/23 69


the point shown.


Object

A B

CD


Worksheet 1A

20/04/23 70

D


point shown.


ObjectA

B

C

Worksheet 2A

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point shown.


ObjectA

B

C

Worksheet 3A

20/04/23 72


D


AB

CObject


point shown.

Worksheet 4A

20/04/23 73

D


AB

CObject


point shown.

Worksheet 5A


20/04/23 74

B

A

D

C

Object


point shown.


Worksheet 6A


20/04/23 75


BA

D C

To enlarge the kite by scale factor x 1/3 from

the point shown.


Object

Worksheet 7A

20/04/23 76

The small rectangle has been enlarged as shown.

Find the centre of enlargement.

Object

A

CD

Image

A/ B/

C/D/

Finding the Centre of Enlargement

B

Draw 2 ray lines through corresponding vertices to locate.


Find Centre 1

20/04/23 77

B/

A/

D/

C/

Image

D

AB

CObject

The small kite has been enlarged as shown. Find

the centre of enlargement.




Find Centre 2

20/04/23 78

Finding the Centre of Enlargement.

B

A

D

C

Object

The large kite has been enlarged by scale

factor x ½ as shown. Find the centre of

enlargement.

A/

B/

D/

C/

Image



Find Centre 3

20/04/23 79



Object

A

CD

Image

A/ B/

C/D/


B

Worksheet 8

20/04/23 80

B/

A/

D/

C/

Image

D

AB

CObject




Worksheet 9

20/04/23 81


B

A

D

C

Object



enlargement.

B/

A/

D/

C/

Image

Workshheet 10

20/04/23 82



Object

A

CD

Image

A/ B/

C/D/


B

Worksheet 8A

20/04/23 83

B/

A/

D/

C/

Image

D

AB

CObject




Worksheet 9A

20/04/23 84


B

A

D

C

Object



enlargement.

A/

B/

D/

C/

Image

Workshheet 10A

20/04/23 85

Enlargement

AA’

Shape A’ is an enlargement of shape A.

The length of each side in shape A’ is 2 × the length of each side in shape A.

We say that shape A has been enlarged by scale factor 2.

20/04/23 86

EnlargementWhen a shape is enlarged the ratios of any length in the image over the corresponding lengths in the original shape (the object) is equal to the scale factor.

A

B

C

A’

B’

C’

= B’C’BC

= A’C’AC

= the scale factorA’B’AB

4 cm6 cm

8 cm

9 cm6 cm

12 cm

64

= 128

= 96

= 1.5

20/04/23 87

Congruence and similarityIs an enlargement congruent to the

original object?

Remember, if two shapes are congruent they are the same shape and size. Corresponding lengths and angles are equal.

In an enlarged shape the corresponding angles are the same but the lengths are different. The object and its image are similar.

Reflections, rotations and translations produce images that are congruent to the original shape.

Enlargements produce images that are similar to the original shape.

20/04/23 88

Enlargement

AA’

Shape A’ is an enlargement of shape A.

The length of each side in shape A’ is 2 × the length of each side in shape A.

We say that shape A has been enlarged by scale factor 2.

20/04/23 89

EnlargementWhen a shape is enlarged the ratios of any length in the image over the corresponding lengths in the original shape (the object) is equal to the scale factor.

A

B

C

A’

B’

C’

= B’C’BC

= A’C’AC

= the scale factorA’B’AB

4 cm6 cm

8 cm

9 cm6 cm

12 cm

64

= 128

= 96

= 1.5

20/04/23 90

Find the scale factor

A

A’

Scale factor = 3

What is the scale factor for the following enlargements?

20/04/23 91

Find the scale factorWhat is the scale factor for the following enlargements?

Scale factor = 2

B

B’

20/04/23 92


Scale factor = 3.5

C

C’


20/04/23 93


Scale factor = 0.5

D

D’


20/04/23 94


A

A’

Scale factor = 3


20/04/23 95

Find the scale factorWhat is the scale factor for the following enlargements?

Scale factor = 2

B

B’

20/04/23 96


Scale factor = 3.5

C

C’


20/04/23 97


Scale factor = 0.5

D

D’


20/04/23 98

Scale factors between 0 and 1What happens when the scale factor for

an enlargement is between 1 and 0?

When the scale factor is between 1 and 0, the enlargement will be smaller than the original object.

Although there is a reduction in size, the transformation is still called an enlargement.

For example,

EE’

Scale factor = 23

20/04/23 99

The centre of enlargementTo define an enlargement we must be given a scale factor and a centre of enlargement.

For example, enlarge triangle ABC by a scale factor of 2 from the centre of enlargement O.

O

A

CB

OA’OA

= OB’OB

= OC’OC

= 2

A’

C’B’

20/04/23 100

The centre of enlargementEnlarge quadrilateral ABCD by a scale factor of from the centre of enlargement O.

O

DA

B C

OA’OA

= OB’OB

= OC’OC

= OD’OE

A’ D’

B’C’

13

= 13

20/04/23 101

This example shows the shape ABCD enlarged by a scale factor of –2 about the centre of enlargement O.

O

A

B

CD

Negative scale factorsWhen the scale factor is negative the enlargement is on the opposite side of the centre of enlargement.

A’

D’

B’

C’

20/04/23 102

Inverse enlargementsAn inverse enlargement maps the image that has been enlarged back onto the original object.

What is the inverse of an enlargement of 0.2 from the point (1, 3)?

In general, the inverse of an enlargement with a scale factor k is an enlargement with a scale factor from the same centre of enlargement.

1k

The inverse of an enlargement of 0.2 from the point (1, 3) is an enlargement of 5 from the point (1, 3).

20/04/23 103

Enlargement on a coordinate grid

20/04/23 104

Finding the centre of enlargementFind the centre the enlargement of A onto A’.

A A’

This is the centre of

enlargement

Draw lines from any two vertices to their images.

Extend the lines until they meet at a point.

20/04/23 105

Finding the centre of enlargementFind the centre the enlargement of A onto A’.

AA’

Draw lines from any two vertices to their images.

When the enlargement is negative the centre of enlargement is at the point where the lines intersect.

20/04/23 106

Describing enlargements

20/04/23 10720/04/23 107

20/04/23 10820/04/23 108

4

2

2

1

For shapes to be similar they must :

1) Have identical angles.

2) Have their sides in the same proportion.

4

2

= 2

2

1

= 2

Two shapes are said to be SIMILAR when one is an ENLARGEMENT of the other.

20/04/23 10920/04/23 109

Match up the PAIRS of Similar Shapes.

Rectangles NOT drawn to scale.1)

2)

3) 4) 5)

6) 7)8)

6

3

8

4

15

3

2

10

15

10

3

2

4

14

2

7

6 ÷ 3 = 2

8 ÷ 4 = 2

3 ÷ 2 = 1.5

15 ÷ 10 = 1.5

10 ÷ 2 = 5

15 ÷ 3 = 5

14 ÷ 4 = 3.5

7 ÷ 2 = 3.5

20/04/23 11020/04/23 110

20/04/23 11120/04/23 Similar Shapes 111

For 2 Triangles to be Similar to each other you only need to check whether or not they have the same angles.

If their angles are the same then their sides will automatically be in the same proportions.

400

400

500

500

Same angles so automatically Similar Triangles.

Menu

20/04/23 11220/04/23 Similar Shapes 112

Match up the PAIRS of Similar Triangles.

Triangles NOT drawn to scale.

Menu

40°80°

40°60°

30°

1) 2)

3)4)

5) 6)

7)

8)

60°

60°

70°

20°

20°

60°70°

60°

50°

20/04/23 11320/04/23 Similar Shapes 113

Menu

20/04/23 11420/04/23 Similar Shapes 114

10 cm

15 cm

Menu

x cm

30 cm

Bob decides to enlarge a poster of himself.

How wide will the enlargement be ?x

30

=10

15

30 × × 30

x = 10 × 30

15

x = 300 15

x = 20 cm

20/04/23 11520/04/23 Similar Shapes 115

Menu

Bob’s work rival decides to reduce the poster so that it is only 3 cm wide. How long will it be ?

x cm

3 cm

20 cm

30 cmx

3

=

30

203 × × 3

x = 30 × 320

x = 9020

x = 4.5 cm

20/04/23 11620/04/23 Similar Shapes 116

Calculate the missing lengths.

{ Each pair of shapes are similar }

Menu

x3

15

9

x

812

10

x8

15

9

5 8x 14

x

2015

9

17 x

8

6 45

130°130°115°

115°

1) 2)

3)4)

5) 6)

5 9.6

16.875

8.75

17.647

12

20/04/23 11720/04/23 Similar Shapes 117

Menu

20/04/23 11820/04/23 Similar Shapes 118

Menu

How high is the church spire ?

1) Hammer a stick into the ground.

2) Line up the top of the stick with the top of the spire. {You will need to put your eye to the ground}

3) We now have 2 Similar Triangles because …

ParallelCommon to both triangles

Both are Right Angles

Corresponding Angles

20/04/23 11920/04/23 Similar Shapes 119

Menu

How high is the church spire ?

4) Measure the height of the stick.

2 m

5) Measure the distances from the ‘eye’ to the stick and the ‘eye’ to the church.

4 m50 m

6) Let the height of the spire be called x.

x

7) You may well find it easier seeing them as two separate triangles

x

50

=

2

4

50 × × 50

x = 2 × 504

x = 25 m

20/04/23 12020/04/23 Similar Shapes 120

Menu

Calculate the missing lengths x

x

x

xx

7

65

15

11

18

17

9

14

5 20

12

6 18

16

1) 2)

3)4)

5.83 9.17

11.7

10.8

Harder Problems

20/04/23 12120/04/23 Similar Shapes 121

Menu

Calculate the missing lengths x

>

>>

>

>

>

x

6

20

5

3x

3

4

12

5x

8

310

6

1) 2)

3)4)A B

C

DE

Prove that triangles ABC and CDE in question 3 are similar.

12.5 1.5

26.7

ACB = DCE (Vertically opposite angles)

CDE = BAC (Alternate angles)

CED = ABC (Alternate angles)

^ ^^ ^^ ^

20/04/23 12220/04/23 Similar Shapes 122

Menu

20/04/23 12320/04/23 Similar Shapes 123

Menu

Each rectangle is enlarged using a Scale Factor = 2

Diagrams not drawn to scale.

1

2

23

5

6

710

2

4

4

6

10

12

14

20

Work out the Areas of each of the rectangles

S.F. = 2

1 × 2 = 2 2 × 4 = 8

2 × 3 = 6 4 × 6 = 24

5 × 6 = 30 10 × 12 = 120

7 × 10 = 70 14 × 20 = 280

In each case how has the Area increased ?

2 8

6 24

30 120

70 280

× 4

× 4

× 4

× 4

20/04/23 12420/04/23 Similar Shapes 124

Menu



1

2

23

5

6

710

3

6

6

9

15

18

21

30


S.F. = 3

1 × 2 = 2 3 × 6 = 18

2 × 3 = 6 6 × 9 = 54

5 × 6 = 30 15 × 18 = 270

7 × 10 = 70 21 × 30 = 630


2 18

6 54

30 270

70 630

× 9

× 9

× 9

× 9

20/04/23 12520/04/23 Similar Shapes 125

Menu



1

2

23

5

6

710

4

8

8

12

20

24

28

40


S.F. = 4

1 × 2 = 2 4 × 8 = 32

2 × 3 = 6 8 × 12 = 96

5 × 6 = 30 20 × 24 = 480

7 × 10 = 70 28 × 40 = 1120


2 32

6 96

30 480

70 1120

× 16

× 16

× 16

× 16

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What is the connection between the Scale Factor and the increase in Area ?

Menu

Scale Factor Increase in Area

( Area multiplier )

2 × 4

3 × 9

4 × 16

( Scale Factor )2 = Increase in Area

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Example 1

Area = 8 cm2

Area = ?

5 cm 15 cmS.F. = 15 ÷ 5 = 3

( Scale Factor )2 = Area multiplier

32 = 9 times

New Area = 8 × 9

= 72 cm2

Example 2 ( Scale Factor )2 = Area multiplier

4 cm x

Area =

10 cm2 Area = 250 cm2

Area multiplier = 250 ÷ 10

= 25 times

Scale Factor = Area multiplierS.F. = 25S.F. = 5

New base : 4 × 5 = 20 cm

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Work out the following : {All of the shapes are Similar}

Area

= 6 cm2

Area = ?

2 cm 8 cm

Area = ?Area

= 45 cm2

12 cm4 cm

Area

= 180 m2Area

= 20 m2

x 5 m 16 m x

Area

= 100 m2Area

= 25 m2

1) 2)

3) 4)

96 cm2 5 cm2

15 m

8 m

20/04/23 12920/04/23 Similar Shapes 129

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20/04/23 13020/04/23 Similar Shapes 130

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Each CUBOID is enlarged using a Scale Factor = 2

Diagrams not drawn to scale.S.F. = 2

211

12

3

2

2

32

34

2

2

42

4

64

4

64

6

8

Work out the Volume of each cuboid.

1 × 1 × 2

= 2

2 × 2 × 4

= 16

1 × 2 × 3

= 6

2 × 4 × 6

= 48

2 × 2 × 3

= 12

4 × 4 × 6

= 96

2 × 3 × 4

= 24

4 × 6 × 8

= 192

In each case how has the Volume increased ?

2 16

6 48

12 96

24 192

× 8

× 8

× 8

× 8

20/04/23 13120/04/23 Similar Shapes 131

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211

12

3

2

2

32

34

3

3

63

6

96

6

96

9

12


1 × 1 × 2

= 2

3 × 3 × 6

= 54

1 × 2 × 3

= 6

3 × 6 × 9

= 162

2 × 2 × 3

= 12

6 × 6 × 9

= 324

2 × 3 × 4

= 24

6 × 9 × 12

= 648


2 54

6 162

12 324

24 648

× 27

× 27

× 27

× 27

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211

12

3

2

2

32

34

4

4

84

8

128

8

128

12

16


1 × 1 × 2

= 2

4 × 4 × 8

= 128

1 × 2 × 3

= 6

4 × 8 × 12

= 384

2 × 2 × 3

= 12

8 × 8 × 12

= 768

2 × 3 × 4

= 24

8 × 12 × 16

= 1536


2 128

6 384

12 768

24 1536

× 64

× 64

× 64

× 64

20/04/23 13320/04/23 Similar Shapes 133

What is the connection between the Scale Factor and the increase in

Volume ?

Menu

Scale Factor Increase in Volume

( Volume multiplier )

2 × 8

3 × 27

4 × 64

( Scale Factor )3 = Increase in Volume

20/04/23 13420/04/23 Similar Shapes 134

( Scale Factor )3 = Volume multiplier

Menu

Example 1

Example 2

Volume

= 50 m3

Volume

= ?5 m 10 m

Scale Factor = 2

23 = 8 times

New Volume = 50 × 8

= 400 m3

( Scale Factor )3 = Volume multiplier

Volume

= 20 m3

Volume

= 540 m3

4 m x

Volume multiplier = 540 ÷ 20

= 27 times

Scale Factor = Volume Multiplier3

S.F. = 273

S.F. = 3New width : 4 × 3 = 12 m

20/04/23 13520/04/23 Similar Shapes 135

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Work out the following : {All of the shapes are Similar}

Volume

= 15 m36

12 Volume

= ?Volume

= 3750 m3Volume

= ?

420

Volume

= 40 m3Volume

= 1080 m3

6x x

36Volume

= 200 m3

Volume

= 12800 m3

1) 2)

3) 4)

120 m3 30 m3

18 m9 m

20/04/23 13620/04/23 Similar Shapes 136

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20/04/23 13720/04/23 Similar Shapes 137

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Shapes are said to be CONGRUENT when they have the same angles and their sides are the same length.

They are identical.

They would fit perfectly over each other.

80°

100°

70°

110° 70° 80°

110° 100° 100°

110°

80°

70°

20/04/23 13820/04/23 Similar Shapes 138

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For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions :

1) If their sides are all the same length then the triangles are identical ( Congruent ).

Side Side Side S.S.S

20/04/23 13920/04/23 Similar Shapes 139

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2) If 2 of their sides are the same length and their INCLUDED angles are the same then the triangles are identical ( Congruent ).

Side Angle Side S.A.S

40° 40°

The INCLUDED angle lies between the 2 pairs of equal length sides.

40° 40°

20/04/23 14020/04/23 Similar Shapes 140

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3) If 2 of their angles are the same and also 1 of their corresponding sides are the same

then the triangles are identical ( Congruent ).

Angle Angle Side A.A.S

40° 40° 70°70°

A corresponding side lies opposite to one of the identical angles.

OR

20/04/23 14120/04/23 Similar Shapes 141


4) If they both have Right angles, they both have the same Hypotenuse and one other side is

the same length then the triangles are identical ( Congruent ).

Right angle Hypotenuse Side R.H.S

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OR

20/04/23 14220/04/23 Similar Shapes 142

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Summary of conditions for Congruent Triangles.

Side Side Side S.S.S Side Angle Side S.A.S

Angle Angle Side A.A.S

Hypotenuse

Hypotenuse

Right angle Hypotenuse Side R.H.S

20/04/23 14320/04/23 Similar Shapes 143

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Which triangles are Congruent to the RED triangle. You must give reasons. ie SSS, AAS, SAS, RHS

Triangles not drawn to scale.

15

9.8 13.280°

60° 40°60° 40°

9.8

13.2

80°

15

15

9.8

13.240°

13.2

15

80° 40°

9.8

1)

2) 3)

4)

5)

A.A.S

S.S.S

6)80°

60° 40°

S.A.S

R.H.S problems

20/04/23 14420/04/23 Similar Shapes 144

Which triangles are Congruent to the RED triangle. You must give reasons. ie SSS, AAS, SAS, RHS

Menu

Triangles not drawn to scale.

3

45

53°

537°

A.A.S

53

R.H.S

53°

5337°

3

5 53° 53 S.A.S or

R.H.S or

A.A.S

37°

20/04/23 145

Enlargement

• Transforms a shape using– A Centre of Enlargement– A Scale Factor

• Exam questions sometimes involve enlargements on an x,y grid

20/04/23 146

Draw a 3 times enlargement of this shape

2

2

1

6

6

3

DIAGONAL lines are best drawn LAST

A shape we change is called an OBJECT

This OBJECT was enlarged with a SCALE FACTOR of 3

The shape it becomes is called the IMAGE

20/04/23 1471 2 3 4 5 6 7 8 9 10 11 12 13 14

10

9

8

7

6

5

4

3

2

1

0

USING A CENTRE OF ENLARGEMENT

Enlarge Triangle by scale factor 2With Centre of Enlargement (0,0)

x

Move one VERTEX at a time

20/04/23 148

After Scale factor 2 Enlargement, how manyTimes will OBJECT fit inside its IMAGE?

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01/11/20151 All of the ship would remain visible as apparent size diminishes. Whereas on a flat Earth