© T Madas

© T Madas

2 shapes which are identical are called:

Congruent

Which transformations produce congruent images?

Congruent shapes have:EqualEqualEqual

lengthsanglesareas

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2 shapes are called similar if :

One is the enlargement of the other

Which transformation produces similar images?

similar shapes have: Equal angles

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The ratio of lengths of similar shapes

is equal to:the scale factor

a A

S.F. = 2

2 shapes are called similar if :

One is the enlargement of the other

Aa

= 2

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6 cm 12 cm

8 c

m

20 cm

The two triangles below are similar.Find their missing sides.

What is the scale factor?

x2

x2

÷2

10 cm

16 c

m

© T Madas

6 cm 18 cm

The two shapes below are similar.Find their missing sides.

What is the scale factor?

x3

7 cm 15 cm

8 c

m

21 cm

24 c

m

5 cm

© T Madas

© T Madas

3

5

x

15

9

4

y 12

5x

2016

3

9 y

6

x

42

49637

10

1612

z

15

y

x

30

24

1820

Calculate the missing lengths in each diagram

figures not to scalemeasurements in cm

© T Madas

7

9

x

54

42

12

y 20

13x

5248

5

9 y

8

x

72

56817

10

1612

z

25

y

x

50

40

3020

Calculate the missing lengths in each diagram

figures not to scalemeasurements in cm

© T Madas

2

4

x

9

4.5 3 y

6

x

18

2799

10

2015

z

16

y

x

40

32

2425

Calculate the missing lengths in each diagram

figures not to scalemeasurements in cm

© T Madas

© T Madas

3

5

x

15

9

4

y 12

5x

2016

3

9 y

6

x

42

49637

10

1612

z

15

y

x

30

24

1820

Calculate the missing lengths in each diagram

figures not to scalemeasurements in cm

© T Madas

7

9

x

54

42

12

y 20

13x

5248

5

9 y

8

x

72

56817

10

1612

z

25

y

x

50

40

3020

Calculate the missing lengths in each diagram

figures not to scalemeasurements in cm

© T Madas

2

4

x

9

4.5 3 y

6

x

18

2799

10

2015

z

16

y

x

40

32

2425

Calculate the missing lengths in each diagram

figures not to scalemeasurements in cm

© T Madas

© T Madas

Angles are equal

Lengths are multiplied by the scale factor

Areas are multiplied by the scale factorsquared

x2 x3

If two shapes are similar:

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Angles are equal

Lengths are multiplied by the scale factor

Areas are multiplied by the scale factorsquared

If two shapes are similar:

S.F. = 2

Lengths: x2Area: x22 = x4

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Angles are equal

Lengths are multiplied by the scale factor

Areas are multiplied by the scale factorsquared

If two shapes are similar:

S.F. = 3

Lengths: x3Area: x32 = x9

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Angles are equal

Lengths are multiplied by the scale factor

Areas are multiplied by the scale factorsquared

If two shapes are similar:

S.F. = 1.5

Lengths:x1.5Area: x1.52 = x2.25

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If 2 shapes are similar:

All lengths are multiplied by the scale factor

Consider a triangle enlarged by a scale factor s

b sb

hsh

= 12

a bh

= 2A as

F o r m a l P r o o f

=A 12

( )sb ( )sh Û

=A 12

2s bh Û=A 2sa

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If 2 shapes are similar:

All lengths are multiplied by the scale factor

F o r m a l P r o o f

Every polygon can be split into triangles so the result holds in general

= 2A as

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© T Madas

A shape with an area of 16 cm2 is enlarged by a scale factor of 4½. Calculate the area of the enlarged shape.

The enlarged shape has an area of 324 cm2

= 2A as

=A 16 .´ 24 5 Û=A 16 .´ 20 25 Û= 324A

Scale factor s = 4.5

© T Madas

© T Madas

A pyramid has a height of 10 cm and a surface area of 200 cm2.A similar pyramid has a height of 25 cm.Find the surface area of the larger pyramid.

The larger pyramid has a surface area of 1250 cm2

= 2A as

=A 200 .´ 22 5 Û=A 200 .´ 6 25 Û=1250A

Scale factor s =2.5

© T Madas

© T Madas

A cylinder has a surface area of 18225 cm2 and a radius of 9 cm.A similar cylinder has a radius of 2 cm.Find the surface area of the smaller cylinder.

The smaller cylinder has a surface area of 900 cm2

= 2A as

=18225 a .´ 24 5 Û=18225 a .´ 20 25 Û= 900a

Scale factor s =4.5

© T Madas

© T Madas

Two similar shapes P and Q have areas of 20 cm2 and 1125 cm2 respectively.Calculate the scale factor from P to Q.

The scale factor from P to Q is 7.5

= 2A as

=1125 20´ 2s Û

.=7 5s

=220s 1125 Û=2s .56 25 Û

© T Madas

© T Madas

Angles are equal

Lengths are multiplied by the scale factor

Areas are multiplied by the scale factorsquared

If two shapes are similar:

Volumes are multiplied by the scale factorcubed

= 2A as

=V vs3

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© T Madas

A cone has volume 100 cm3 and height of 10 cm.A similar cone has a height of 15 cm.Calculate the volume of the larger cone.

The larger cone has a volume of 337.5 cm3

= 3V vs

=V 100 .´ 31 5 Û=V 100 .´ 3 375 Û

.= 337 5V

Scale factor s =1.5

© T Madas

© T Madas

A prism has a height of 10 cm.A similar prism has a height of 20 cm and a volume of 1000 cm3.Calculate the volume of the smaller prism.

The smaller prism has a volume of 125 cm3

= 3V vs

=1000 v ´ 32 Û=1000 8v Û=125v

Scale factor s =2

© T Madas