Unit 19 Section 2 : Similar shapes

Similar shapes are those which are enlargements of each other; for example, the three triangles shown below are similar:
For example, to get triangle A from triangle C we enlarge it by multiplying all the sides by a scale factor of 2.
Similarly, we can get triangle C from triangle A by multiplying each side by a scale factor of ½.
Triangles A and C are therefore enlargements of each other, and as such are similar.

Finding the scale factor
Often we will want to find the scale factor needed to between two shapes. There is a simple way to do this.

Firstly you need to find a corresponding side in each shape where you know the length of both.
Next you need to divide the length in the shape you are going to by the length in the shape you are coming from.

For example, in triangles A and B above, the bottom sides 6cm and 9cm correspond to each other.

To find the scale factor from A to B, we divide the length in B by the length in A.
You can see that 9 ÷ 6 = 1.5, so the scale factor from A to B is 1.5.
Similarly, to find the scale factor from B to A, we divide the length in A by the length in B.
Now we can see that 6 ÷ 9 = 2/3 so the scale factor from B to A is 2/3.

Practice Question

The following diagram shows two similar shapes:
Work out the answer to each of these questions then click on the button marked Click on this button below to see the correct answer to see whether you are correct.

(a) What is the scale factor from Triangle 1 to Triangle 2?

(b) What is the length of side DF?

(c) What is the scale factor from Triangle 2 to Triangle 1?

(d) What is the length of side BC?

 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

Question 1
The following diagram shows two similar rectangles:
(a) What is the scale factor from rectangle EFGH to rectangle ABCD?

(b) What is the length of side CD?
cm
Question 2
The following diagram shows two similar triangles:
(a) What is the scale factor from triangle ABC to triangle DEF?
cm
(b) What is the length of side EF?
cm
(c) What is the length of side AB?
cm
Question 3
Two similar isosceles triangles are shown in the diagram below:
(a) What is the length of side DE?
cm
(b) What is the length of side AC?
cm
(c) What is the scale factor from triangle DEF to triangle ABC?

(d) What is the length of side BC?
cm
Question 4
The following diagram shows two similar triangles:
(a) What is the length of side GE?
cm
(b) What is the length of side FG?
cm
Question 5
The following diagram shows three similar triangles:
(a) What is the length of side EG?
cm
(b) What is the length of side HJ?
cm
(c) What is the length of side EF?
cm
(d) What is the length of side AB?
cm
Question 6
The following diagram shows three similar triangles:
Note that the triangles have also been rotated, so you will need to be careful working out which sides correspond.

(a) What is the length of side HI?
cm
(b) What is the length of side BC?
cm
(c) What is the length of side AC?
cm
(d) What is the length of side DF?
cm

Question 7
The following diagram shows two similar shapes:
The length of the side AB is 6cm and the length of the side IJ is 4 cm.

(a) If AH = 12cm, calculate the length IP.
cm
(b) If BC = 3 cm, calculate the length JK.
cm
(c) If DE = B C, determine the length LM.
cm
(d) Calculate the length FG
cm
(e) Calculate the length NO.
cm
(f) If MN = 3 cm, determine the length EF.
cm

Question 8
In the diagram shown below the lines BE and CD are parallel. This means that triangles ABE and ACD are similar.
The length of AB is 4.4cm and the length of AD is 13.5cm.

(a) What is the length of side AC?
cm
(b) What is the length of side BC?
cm
(c) What is the length of side AE?
cm
(d) What is the length of side DE?
cm


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Produced by A.J.Reynolds August 2008