Transcript Year 9 Similarity - DrFrostMaths.com

GCSE Similarity
Dr J Frost ([email protected])
GCSE Revision Pack References: 131, 137, 171, 172
Last modified: 19th February 2015
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 metres
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
b
(flipping is again allowed)
b
a
a
a
b?
Similarity
These two triangles are
similar. What is the
missing length, and why?
5
7.5
?
8
12
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.:
5
𝑥
=
8 12
12
Find scale factor: 8
Then multiply or divide other sides by
scale factor as appropriate.
12
𝑥 =5×
8
Quickfire Examples
Given that the shapes are similar, find the missing side (the first 3 can be done in your head).
1
2
12
?
10
32
?
24
18
15
15
20
4
3
17
24
11
20
25
?
40
30
25.88
?
Harder Problems
1
In the diagram BCD is similar to triangle
ACE. Work out the length of BD.
Work out with your neighbour.
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)
𝐵𝐷 7.5
=
4
10
?→
𝐵𝐷 = 3
Since EC = 12cm, by
Pythagoras, DC = 9cm. Using
similar triangles AEF and CDE:
15 ? 𝐸𝐹
=
9
12
Thus 𝐸𝐹 = 20
(Vote with your diaries) What is the length x?
1
x
8
4
8
9
10
12
(Vote with your diaries) What is the length x?
4
9
5
8
x
6
6.5
6.66...
(Vote with your diaries) What is the length x?
x
7.5
15
10
5
11.25
3
6.5
Exercise 1
7
1
5𝑐𝑚
𝑟
5
2
𝑦
𝑥
12𝑐𝑚
15𝑚
10𝑐𝑚
9𝑐𝑚
?
5
6
5
?
𝑎
𝑏
𝐻
𝑎
𝑂
ℎ
𝑏
𝐵
N2
3
?
𝑥 = 1.5
𝒂𝒉
𝒃
By similar triangles 𝑨𝑯 =
Using Pythag on 𝚫𝑨𝑶𝑯:
𝒂𝟐 𝒉𝟐
𝟐
𝟐
𝒂 =𝒉 + 𝟐
𝒃
Divide by 𝒂𝟐 𝒉𝟐 and we’re done.
1.8𝑚
?
𝑥 = 10.8
5
𝑥
?
ℎ
𝑥
N1
Let 𝑎 and 𝑏 be the lengths of the two shorter
sides of a right-angled triangle, and let ℎ be the
distance from the right angle to the hypotenuse.
1
1
1
Prove 2 + 2 = 2
𝐴
?
?
𝑩𝑪 = 𝟖𝒄𝒎
𝑨𝑪 = 𝟏𝟐. 𝟓𝒄𝒎
7
𝑥 = 4.2
𝐶
6
3
4
𝑥
𝐵
𝑥 = 5.25
𝑦 = 5.6
?
N3
1.2𝑚
3.7𝑚
𝑟 = 3.75𝑐𝑚
8
A swimming pool is filled with
water. Find 𝑥.
4
3𝑐𝑚
4
3.75
12𝑐𝑚
3
2𝑐𝑚
𝐴
[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 𝑃 to area 𝑄?
4
[Source: IMO] A square is
inscribed in a 3-4-5 right-angled
triangle as shown. What is the
side-length of the square?
The two unlabelled triangles
Suppose the length of the
𝟑−𝒙
square is 𝒙. Then
=
𝒙
.
𝟒−𝒙
?
Solving: 𝒙 =
𝒙
𝟏𝟐
𝟕
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.
𝑷: 𝑸 = 𝟔: 𝟏𝟓
?
A4/A3/A2 paper
A4
A5
𝑦
A5
𝑥
“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?
𝑥 2𝑦
=
𝑦
𝑥
∴ 𝑥?
= 2𝑦
So the length is 2 times
greater than the width.
Scaling areas and volumes
A Savvy-Triangle is enlarged by a scale factor of 3 to form a Yusutriangle.
2cm
6cm
?
3cm
9cm
?
?2
Area = 3cm
Area = 27cm
? 2
Length increased by a factor of 3?
Area increased by a factor of 9?
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 3m2. What is the area of shape Y?
Shape X
Shape Y
Bro Tip: This is my
own way of working
out questions like
this. You really can’t
go wrong with this
method!
×5
Length:
Area:
3m2
×?25
75m
? 2
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 A
Length:
Area:
5cm
30cm2
Shape B
×?2
×?4
?
10cm
120cm2
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 30cm2
and the surface area of B is 270cm2. If the volume of shape A is 80cm3, what is the
volume of shape B?
Shape A
Shape B
×?3
Length:
Area:
30cm2
Volume:
80cm3
×?9
×?27
270cm2
2160cm
? 3
Test Your Understanding
These 3D shapes are mathematically similar.
If the surface area of solid A is 20cm2. What is
the surface area of solid B?
B
A
Volume = 10cm3
Volume = 640cm3
Solid A
Solid B
×4
Length:
Area:
20cm2
Volume:
10cm3
× 16
× 64
?
320cm2
640cm3
Answer = 320cm2
Exercises
1
Copy the table and determine
the missing values.
Shape A
Length:
Area:
Volume:
2
Shape B
×2
3cm
5cm2
10cm3
?
?
×4
×8
Shape A
3
?
?
Shape B
?
?
?
×3
5m
8m2
12m3
×9
× 27
Length:
Area:
Volume:
6
?
?
15m
72m2
324m3
?
?
?
× 25
?
× 125
?
5cm
100cm2
375cm3
Determine the missing values.
Shape A
Length:
Area:
Volume:
?
6m
8m2
10cm3
Shape B
?
?
?
× 1.5
× 2.25
× 3.375
9m
18m2
33.75cm3
?
[2007] Two cones, P and Q, are mathematically
similar. The total surface area of cone P is 24cm2.
The total surface area of cone Q is 96cm2.
The height of cone P is 4 cm.
(a) Work out the height of cone Q.
𝟗𝟔 ÷ 𝟐𝟒 = 𝟐
𝟒 × 𝟐 = 𝟖𝒄𝒎
(b) The volume of cone P is 12 cm3. Work out the
volume of cone Q.
𝟏𝟐 × 𝟐𝟑 = 𝟗𝟔𝒄𝒎𝟑
?
Shape B
×5
1cm
4cm2
3cm3
[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 80cm3.
Calculate the volume of cylinder B.
𝟖 × 𝟏. 𝟓𝟑 = 𝟐𝟕𝟎𝒄𝒎𝟑
?
Determine the missing values.
Shape A
4
6cm
20cm2
80cm3
Determine the missing values.
Length:
Area:
Volume:
5
?
7
The surface area of shapes A and B are 𝑥 and 𝑦
respectively. Given that the length of shape B is 𝑧,
write an expression (in terms of 𝑥, 𝑦 and 𝑧) for
the length of shape A.
𝒚
𝒛 𝒙
𝒛÷
→
𝒙
𝒚
?
Test Your Understanding
Bro Hint: Scaling mass is the same as
scaling what? Volume
?
50
25
9
25
5
=
9
3
Scale factor of area: 18 =
Scale factor of length:
?
Scale factor of volume/mass:
500 ÷
5 3
3
125
= 𝟏𝟎𝟖𝒈
27
=
125
27
Units of Area and Volume
We can use the same principle to find how to convert between units of volume and area.
1m
100cm
1m
100cm
𝑨𝒓𝒆𝒂 = 𝟏 ? 𝒎𝟐
𝑨𝒓𝒆𝒂 = 𝟏𝟎 ?
𝟎𝟎𝟎 𝒎𝟐
Example:
What is 8.3m2 in cm2?
2
× 100
?
8𝑚2
83 ?
000 𝑐𝑚2
Quickfire Questions
1
What is 42cm2 in mm2?
42𝑐𝑚
2
2
× 10
?2
4?200 𝑚𝑚
5
2
What is 2m2 in mm2?
2𝑚2
?2
× 1000
What is 5.1cm2 in mm2?
5.1𝑚
6
2 000
? 000 𝑚𝑚2
2
× 10
?2
510𝑚𝑚2
?
What is 2km3 in m3?
2𝑘𝑚3
× 1000
? 3
2 000 000 000 𝑚3
?
3
What is 3m3 in cm3?
3𝑚2
4
?3
× 100
7
3 000
? 000 𝑚𝑚2
What is 13cm3 in mm3?
13𝑚2
× 10?3
13?000 𝑚𝑚2
What is 4.25m2 in mm3?
4.25𝑚2
8
?3
× 1000
4 250 000
? 000 𝑚𝑚2
What is 10.01km2 in mm2?
2
× 1000000
?
10.01𝑘𝑚2
10 010 000
? 000 000 𝑚𝑚2