Similarity Dr J Frost jfrosttiffinkingstonschuk wwwdrfrostmathscom Last modified 31 st August 2015 GCSE Revision Pack References 131 137 171 172 GCSE Specification Pack Ref ID: 620810
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Slide1
GCSE Similarity
Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com
Last modified:
31
st August 2015
GCSE Revision Pack References: 131, 137, 171, 172Slide2
GCSE Specification
Pack Ref
Description
171
Solve problems involving finding lengths in similar shapes.
172
Understand the effect of enlargement for perimeter, area and volume of shapes and solids. Know the relationships between linear, area and volume scale factors of mathematically similar shapes and solids
131
Convert between units of area
137
Convert between volume measures, including cubic centimetres and cubic metresSlide3
Similarity
vs
Congruence
Two shapes are congruent if:
They are the same
shape and size
(flipping is allowed)
Two shapes are similar if:
They are the same
shape
(flipping is again allowed)
a
a
a
b
b
b
?
?
!
!Slide4
Similarity
5
8
12
7.5
These two triangles are similar. What is the missing length, and why?
?
There’s two ways we could solve this:
The ratio of the left side and bottom side is the same in both cases, i.e.:
Find scale factor:
Then multiply or divide other sides by scale factor as appropriate.
Slide5
Quickfire
Examples
Given that the shapes are similar, find the missing side (the first 3 can be done in your head).
10
15
18
12
?
15
20
24
32
?
20
24
30
25
?
1
2
3
4
17
40
11
25.88
?Slide6
Harder Problems
In the diagram BCD is similar to triangle ACE. Work out the length of BD.
?
1
2
The diagram shows a square inside a triangle. DEF is a straight line.
What is length EF?
(Hint: you’ll need to use
Pythag
at some point)
Since EC = 12cm, by Pythagoras, DC = 9cm. Using similar triangles AEF and CDE:
Thus
Work out with your neighbour.
?Slide7
9
8
12
10
(Vote with your diaries) What is the length
x
?
x
1
4
8Slide8
6
5
6.66...
6.5
(Vote with your diaries) What is the length
x
?
x
4
8
9Slide9
11.25
5
6.5
3
(Vote with your diaries) What is the length
x
?
x
7.5
10
15Slide10
Exercise 1
A swimming pool is filled with water. Find
.
4
1
2
5
6
?
?
?
?
?
3
?
?
N
1
[Source: IMO] A square is inscribed in a 3-4-5 right-angled triangle as shown. What is the side-length of the square?
Suppose the length of the square is
. Then
.
Solving:
N
2
Let
and
be the lengths of the two shorter sides of a right-angled triangle, and let
be the distance from the right angle to the hypotenuse. Prove
N
3
By similar triangles
Using
Pythag
on
Divide by
and we’re done.
?
?
[Source: IMC] The
diagram shows a square, a diagonal and a line joining a vertex to the midpoint of a side. What is the ratio of area
to area
?
The two unlabelled triangles are similar, with bases in the ratio 2:1. If we made the sides of the square say 6, then the areas of the four triangles are 12, 15, 6, 3.
?Slide11
A4/A3/A2 paper
A4
A5
A5
“A” sizes of paper (A4, A3, etc.) have the special property that what two sheets of one size paper are put together, the combined sheet is mathematically similar to each individual sheet.
What therefore is the ratio of length to width?
So the length is
times greater than the width.
?Slide12
Scaling areas and volumes
3cm
2cm
6cm
9cm
A Savvy-Triangle is enlarged by a
scale factor of 3
to form a
Yusutriangle
.
Area = 3cm
2
Area = 27cm
2
Length increased by a factor of 3
Area increased by a factor of 9
?
?
?
?
?
?Slide13
Scaling areas and volumes
For area, the scale factor is
squared
.
For volume, the scale factor is
cubed
.
Example:
A shape X is enlarged by a scale factor of 5 to produce a shape Y. The area of shape X is 3m
2
. What is the area of shape Y?
Bro Tip
: This is my own way of working out questions like this. You really can’t go wrong with
this method!
Shape X
Shape Y
Length:Area:
3m
2
75m2
?
?Example: Shape A is enlarged to form shape B. The surface area of shape A is 30cm2
and the surface area of B is 120cm2. If shape A has length 5cm, what length does shape B have?Shape AShape BLength:
Area:
30cm
2
120cm2
?
5cm
10cm
?
?Slide14
2160cm
3
Scaling areas and volumes
For area, the scale factor is
squared
.
For volume, the scale factor is
cubed
.
Example 3:
Shape A is enlarged to form shape B. The surface area of shape A is 30cm
2
and the surface area of B is 270cm
2
. If the volume of shape A is 80cm
3, what is the volume of shape B?Shape A
Shape BLength:Area:Volume:
30cm
2
270cm2
?
?
?
80cm3
?Slide15
Test Your Understanding
A
B
These 3D shapes are mathematically similar.
If the surface area of solid A is 20cm
2
. What is the surface area of solid B?
Volume = 10cm
3
Volume = 640cm
3
640cm
3
Solid A
Solid B
Length:
Area:
Volume:
20cm
2
320cm2
10cm
3Answer = 320cm2?Slide16
Exercises
1
Copy the table and determine the missing values.
Shape A Shape B
Length:
Area:
Volume:
3cm
5cm
2
10cm
3
6cm
20cm
2
80cm
3
?
?
?
?
2
Determine the missing values.Shape A Shape B
Length:
Area:Volume:
5m
8m
212m3
15m
72m
2
324m
3
?
?
?
?
3
Determine the missing values.
Shape A Shape B
Length:
Area:
Volume:
1cm
4cm
2
3cm
3
5cm
100cm
2
375cm
3
?
?
4
Determine the missing values.
Shape A Shape B
Length:
Area:
Volume:
6m
8
m
2
10cm
3
9m
18m
2
33.75cm
3
?
?
?
?
?
?
?
?
?
5
[2003] Cylinder A and cylinder B are mathematically similar. The
length of cylinder A is 4 cm and the length of cylinder B is 6 cm.
The volume of cylinder A is
80cm
3
.
Calculate
the volume of cylinder B.
[2007
]
Two
cones, P and Q, are mathematically
similar. The
total surface area of cone P is
24cm
2
.
The total surface area of cone Q is
96cm
2
.
The height of cone P is 4 cm.
(
a) Work
out the height of cone Q
.
(b) The
volume of cone P is 12
cm
3
. Work
out the volume of cone Q
.
6
7
The surface area of shapes A and B are
and
respectively. Given that the length of shape B is
, write an expression (in terms of
,
and
) for the length of shape A.
?
?
?
?Slide17
Test Your Understanding
Bro Hint:
Scaling mass is the same as scaling what?
Volume
?
Scale factor of area:
Scale factor of length:
Scale factor of volume/mass:
?Slide18
Test Your Understanding
Nov 2014
NonCalc
?
?Slide19
Units of Area and Volume
We can use the same principle to find how to convert between units of volume and area.
1m
1m
100cm
100cm
?
?
Example:
What is 8.3m
2
in cm
2
?
?
?Slide20
Quickfire
Questions
What is 42cm
2
in mm
2
?
?
?
What is 2m
2
in mm
2
?
?
?
What is 3m
3
in cm
3
?
?
?
What is 13cm
3
in mm
3
?
?
?
1
2
3
4
What is 5.1cm
2
in mm
2
?
?
?
What is 2km
3
in m
3
?
?
?
What is 4.25m
3
in mm
3
?
?
?
What is 10.01km
2
in mm
2
?
?
?
5
6
7
8