GCSE Ratio and Similar Triangles

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GCSE
Ratio, Scale and Similar Triangles
Dr J Frost ([email protected])
Last modified: 22nd February 2014
RECAP: Ratio
Most ratio problems come in three flavours:
1
Total given.
3
The ratio of cats to dogs in
Battersea Dogs Home is 3:5. There
are 240 animals in total. How many
cats are there?
8 parts
1 part
3 parts
2
= 240
= 30 ?
= 90
Alternately use a
fractional approach.
3/8 of the animals
are cats. 3/8 of 240
= 90
One quantity given
The ratio of boys to girls in a class is
5:6. There are 15 boys. How many
girls are there?
5 parts = 15
1 part ? = 3
6 parts = 18
Difference given
Some pocket money is shared
between Alice and Bob in the ratio
3:7. Bob gets £16 more than Alice.
What was the total amount of
pocket money shared?
4 parts = £16
1 part ?= £4
10 parts = £40
Test Your Understanding
1 [Edexcel Nov 2012] Talil is going to make
some concrete mix. He needs to mix
cement, sand and gravel in the ratio
1 : 3 : 5 by weight.
Talil wants to make 180 kg of concrete
mix.
Talil has
15 kg of cement
85 kg of sand
100 kg of gravel
Does Talil have enough cement, sand and
gravel to make the concrete mix?
9 parts = 180kg
1 part (cement) = 20kg
3 parts (sand)
? = 60kg
5 parts (gravel) = 100kg
Insufficient cement.
2 I need to make 300ml of squash, which
consists of 1 part concentrate to 5 parts
water. How much water do I use?
6 parts = 300ml
1 part = 50ml
?
5 parts = 250ml
3 The ratio of red to blue marbles in a bag
is 7:4. If there are 49 red marbles, how
many blue marbles are there?
7 parts = 49
1 part = 7?
4 parts = 28
4 Frostonia has a population 10,000 more
than Yusufland. If the ratio of people in
Frostonia to Yusufland is 11:7, what is
the population of Yusufland?
4 parts = 10,000
1 part = 2,500
?
7 parts = 17,500
RECAP: Map Scale
Edexcel IGCSE Jan 2014
Method 1:
Method 2:
14cm is 14/4 = 3.5 times bigger. ? 4cm : 1km so 1cm : 0.25km
1km x 3.5 = 3.5km
So 14cm : 3.5km
1km = 1000 x 100 = 100,000cm
100,000 / 4 = 25,000
?
So ratio is 1 : 25,000
Test Your Understanding
1 A map scale is 1 : 10 000. Two towns
are 14cm apart on the map. What
actual distance does this represent in
km?
1.4km ?
2 5cm on a map represents 18km in real
life.
a) What does 8cm represent?
28.8km ?
b) What is the map scale in the form
1:n?
1: 360 000?
3 16cm on a map represents 12km in
real life.
a) What does 20cm represent?
15km ?
b) What is the map scale in the form
1:n?
1: 75 000?
4 A map scale is 1 : 50 000. If a
supermarket is 3.2km away in real
life, how close will it appear on the
map?
6.4cm ?
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

=
8 12
We scale each length by the
same amount. i.e.:
 12
=
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
?
(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
Harder Problems
1
In the diagram BCD is similar to triangle
ACE. Work out the length of BD.
(both these questions were on past Year 9 landmarks)
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)
 7.5
=
4
10
?→
 = 3
Since EC = 12cm, by
Pythagoras, DC = 9cm. Using
similar triangles AEF and CDE:
15 ? 
=
9
12
Thus  = 20
Harder Problems
3
4
A square is inscribed in a 3-4-5 rightangled triangle as shown. What is the
side-length of the square?
5
3
Work out the length of:
4
a) 
b) 
2
3
12
7

12
=

3
= 12 so  = 2.5. So  = 12.5
so  = 8
10
?
Many students didn’t realise that the triangle on the
left gets flipped upside down on the right, hence got
an incorrect answer of 13.6 for .
Suppose the length
? of the
3−

square is . Then  = 4−.
Then solve.
Exercises
7
A swimming pool is filled with
water. Find .
1
2
15
5
3
5

4
3.75
12


1.2
9
3.7

?
 = 3.75
1.8
?
 = 10.8
5
4
8
6
3
4
5


7
?
 = 4.2
?
 = 1.5
 = 5.25
 = 5.6
?
A4/A3/A2 paper
A4
A3

A3

“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?
 2
=


∴ ?
= 2
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.
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.
 ÷  = 
 ×  = 
(b) The volume of cone P is 12 cm3. Work out the
volume of cone Q.
 ×  = 
?
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.
 × .  = 
?
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  and 
respectively. Given that the length of shape B is ,
write an expression (in terms of ,  and ) for
the length of shape A.

 
÷
→


?
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
= 
27
=
125
27

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