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
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 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 Test
There are 3 methods to test whether 2 triangles are similar.
1. 2 pairs of corresponding angles are equal
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
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)
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)
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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