CAIE IGCSE Mathematics Extended Similarity Questions

Practise proving shapes similar and applying linear, area and volume scale factors to lengths, areas, capacities and related geometric problems.

Exam points

justify triangle similarity with two equal angle pairs or valid parallel-line angle facts
use the square of the length scale factor for area and its cube for volume or capacity
match corresponding measurements and select the correct root when recovering a length scale factor

Covered syllabus points

E4.4.1 Calculate lengths of similar shapes.E4.4.2 Use the relationships between lengths and areas of similar shapes and E4.4.3 Solve problems and give simple explanations involving similarity

[Maximum number: 7]

NOT TO SCALE

In the diagram, S lies on PQ and T lies on PR.
ST is parallel to QR.

(a)

Explain why triangle PQR is mathematically similar to triangle PST.

Give a geometrical reason for each statement you make.

[ 3 ]

Three correct pairs with correct reasons

$\angle PQR=\angle PST$ and corresponding angles
$\angle PRQ=\angle PTS$ and corresponding angles
$\angle QPR=\angle TPS$ and common angle
OR
Two correct pairs with correct reasons and correct similarity condition, e.g. AAA

B2 for two correct pairs with correct reasons
or B1 for one correct pair with correct reason

(b)

$S T=3 \mathrm{~cm}, Q R=9 \mathrm{~cm}$ and $P S=5 \mathrm{~cm}$ .

Work out PQ.

[ 2 ]

15

M1 for $\frac39=\frac5{PQ}$ oe or better

(c)

The area of triangle PST is $2 k \mathrm{~cm}^{2}$ .

Find, in terms of k, the area of quadrilateral QRTS. $\mathrm{cm}^{2}$

[ 2 ]

16k

M1 for $3^2$ or $(\frac13)^2$ oe

[Maximum number: 3]

A sector of a circle has radius 8 cm and area 160 cm^2.

A mathematically similar sector has radius 20 cm .
Calculate the area of the larger sector.
cm^2
[3]

1000 cao
M2 for $160\times\left(\frac{20}{8}\right)^2$ or $160\div\left(\frac{8}{20}\right)^2$ oe
or M1 for $\left(\frac{20}{8}\right)^2$ or $\left(\frac{8}{20}\right)^2$ oe

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Covered syllabus points

E4.4 Similarity question 1

Question (a)

Question (b)

Question (c)

E4.4 Similarity question 2