Middle Years

7.07 Similar triangle tests

Worksheet
Similar triangles
1

State whether the following pairs of triangles are similar, and if so, state the similarity test that proves this.

a
b
c
d
e
f
g
h
2

For each of the following sets of triangles:

i

Identify which two are similar.

ii

Identify the similarity test that the pair satisfy: AAA, SSS, SAS, or RHS.

a
A
B
C
D
b
A
B
C
D
c
A
B
C
D
d
A
B
C
D
e
A
B
C
D
f
A
B
C
D
g
A
B
C
D
h
A
B
C
D
Applications
3

Consider the following diagram:

a

Identify two similar triangles and state the similarity test that they satisfy.

b

Given that DE = 7, what is the length of BC?

4

Consider the following diagram:

a

State the similarity test that triangles \triangle ABX and \triangle CDX satisfy.

b

Given that AB = 5, what is the length of CD?

5

Consider the following diagram:

a

Identify two similar triangles and state the similarity test that they satisfy.

b

Given that KL = 20, what is the length of MN?

6

Consider the following diagram, where AD is a straight line:

What information is required to know that triangles \triangle ABC and \triangle DBE are in fact similar?

7

Consider the following diagram, where EH and FG are straight lines:

What information is required to know that triangles \triangle EFX and \triangle GHX are in fact similar?

8

Consider the following diagram, where TZ and UY are non-parallel straight lines:

a

Explain why the triangles \triangle TUX and \triangle YZX are not similar.

b
What length does YZ need to be for the two triangles to be similar?
9

Peter constructs a ramp in the shape of a right-angled triangle. To give the ramp more strength, he wants to place a brace from the base point across to the hypotenuse. The brace should meet the hypotenuse at a right angle, as shown in the diagram:

a

State the similarity test that the triangles \triangle ABC and \triangle ADB satisfy.

b

How long does the brace, BD, need to be?

c

Hence, find the length x.

10

A horizontal brace lies between the slanted sides of an A-shaped ladder. In this position, the top of the ladder is 220 \text{ cm} from the ground. The brace is 44 \text{ cm} in length, and sits 71 \text{ cm} from the top of the ladder.

How far apart are the feet of the ladder? Round your answer to the nearest whole centimetre.

11

A large tree casts a shadow of 30\text{ m}. At the same time, a 2\text{ m} high stick, standing vertically, casts a shadow of 3\text{ m}.

a

State the reasons why the triangles \triangle ABE and \triangle DCE are similar.

b

If the height of the tree is h metres, find h.