year 8 maths

500 Chapter 8 Transformations and congruence
8G Similar triangles
EXTENSION
Finding the approximate height of a tree or the width of a gorge using only simple equipment is possible
without actually measuring the distance directly. Similar triangles can be used to calculate distances
without the need to climb the tree or cross the gorge. It is important, however, to ensure that if the
mathematics of similar triangles is going to be used, then the two triangles are in fact similar. We learnt
earlier that there were four tests that help to determine if two triangles are congruent. Similarly there
are four tests that help establish whether or not triangles are similar. Not all side lengths or angles are
required to prove that two triangles are similar.
Let’s start: How much information is enough?
Given a certain amount of information, it may be possible to draw two triangles that are guaranteed to be
similar in shape.
Decide if the information given is enough to guarantee that the two triangles will always be similar. If
you can draw two triangles ( △ABC and △DEF ) that are not similar, then there is not enough information
provided.
• ∠A = 30° and ∠D = 30°
• ∠A = 30°, ∠B = 80° and ∠D = 30°, ∠E = 80°
• AB = 3 cm, BC = 4 cm and DE = 6 cm, EF = 8 cm
• AB = 3 cm, BC = 4 cm, AC = 5 cm and DE = 6 cm, EF = 8 cm, DF = 10 cm
• AB = 3 cm, ∠A = 30° and DE = 3 cm, ∠D = 30°
• AB = 3 cm, AC = 5 cm, ∠A = 30° and DE = 6 cm, DF = 10 cm, ∠D = 30°
Key ideas
■ Two triangles are similar when:
• Two angles of a triangle are equal to two angles of another triangle.
85°
60°
60°
85°
• Three sides of a triangle are proportional to three sides of another triangle.
C
2 m
A
3 m
1 m
B
F
4 m
D
6 m
2 m
E
DE
AB = EF
BC = DF
AC = 2
Scale factor
• Two sides of a triangle are proportional to two sides of another triangle, and the included
angles are equal.
B C E F DE
AB = DF
AC = 1.5
2 cm
3 cm
Scale factor
40° 3 cm
4.5 cm
40°
A
D
Cambridge Maths NSW
Stage 4 Year 8 Second edition
ISBN 978-1-108-46627-1 © Palmer et al. 2018
Cambridge University Press
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