Similar Triangles Test – O Level Exam Preparation Guide

In this Similar Triangles revision note, you will learn the properties of Similar Triangles and how to test for them.

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similar triangles, o level math notes

 

Similar Triangles

When two triangles are similar, they have the same shape but different size.

When a triangle undergoes an enlargement or reduction, the shape remains the same but the sizes are different. Hence they are similar triangles.

similar triangles

Similar triangles Properties

ΔABC is similar to ΔXYZ (Note the order)

if and only if…

the ratio of corresponding sides are equal,

AB/XY = AC/XZ =  BC/YZ

and corresponding angles are equal.

∠A = ∠X , ∠B = ∠Y and ∠C = ∠Z

similar triangles

Similar Triangles Test

There are 3 methods to test whether 2 triangles are similar.

1. 2 pairs of corresponding angles are equal

similar triangles test for same angles

i.e. ∠A = ∠D, ∠B = ∠E (AA)

∆ABC and ∆DEF are similar.

Note: When 2 angles are the same, the 3rd angle will be the same too because the sum of angles is always 180 degrees

similar triangles test for same ratios

In ∆ABC and ∆DEC,

∠ACB = ∠DCE (common)

∠ABC = ∠DEC (corresponding angles)

∠BAC = ∠EDC (corresponding angles)

∴ ∆ABC is similar to ∆DEC. (AA Similarity)

 

2. 3 pairs of corresponding sides are in the same ratio

i.e. AB/DE =  BC/EF =  AC/DF (SSS)

similar triangles test

In ∆ABC and ∆DEF,

AB/DE = 12/18 = 2/3, BC/EF = 8/12 = 2/3, AC/DF = 10/15 = 2/3

∴ ∆ABC is similar to ∆DEF. (SSS Similarity)

 

3. 2 pairs of corresponding sides are in the same ratio and a pair of included angles is equal

i.e. AB/DE = AC/DF and ∠A = ∠D (SAS)

similar triangles example

In ∆ABC and ∆DEF,

∠BAC = ∠EDF = 50°

AB/DE = 3/5,  AC/DF = 12/20 = 3/5

∴ ∆ABC is similar to ∆DEF (SAS Similarity)

Similar Triangles Example

In the diagram below, PQ is parallel to BC. APB and AQC are straight lines. Given that AP = x cm, BC = 18 cm, PB = 6 cm and PQ = 10 cm, find
(a) x.
(b) the ratio of AQ : AC.

Solution

Angle A is a common angle, Angle APQ = Angle ABC (corresponding angle).

∆APQ and ∆ABC are similar triangles (AA Similarity).

Therefore, the ratios of the corresponding sides are equal.

AP/AB =PQ/BC = AQ/AC

 

(a) To find x

AP/AB = PQ/BC

x/(x + 6) cm = 10cm/18cm

10(x + 6) = 18 x

10x + 60 = 18 x

8 x = 60

x = 7.5 cm

 

(b) To find the ratio of AQ : AC

Since the triangles are similar, the ratio of AQ : AC is the same as its corresponding sides PQ: BC.

AQ/ AC = PQ/ BC = 10 cm/18 cm = 5 : 9

 

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