© t madas. 2 shapes which are identical are called: congruent which transformations produce...
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© T Madas
© T Madas
2 shapes which are identical are called:
Congruent
Which transformations produce congruent images?
Congruent shapes have:EqualEqualEqual
lengthsanglesareas
© T Madas
2 shapes are called similar if :
One is the enlargement of the other
Which transformation produces similar images?
similar shapes have: Equal angles
© T Madas
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
© T Madas
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:
© T Madas
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
© T Madas
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
© T Madas
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
© T Madas
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
© T Madas
© 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
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© 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