CAIE IGCSE Mathematics Extended Geometry Question Bank

Practise geometric language, constructions, bearings, similarity, symmetry, angle properties, polygons and circle theorems across CAIE IGCSE.

Covered syllabus points

E4.1 Geometrical terms E4.2 Geometrical constructions E4.3 Scale drawings E4.4 Similarity E4.5 Symmetry E4.6 Angles E4.7 Circle theorems I E4.8 Circle theorems II

[Maximum number: 6]

(a)

The scale drawing shows two sides, AB and BC, of a field. The scale is 5 centimetres represents 200 metres.

Figure for Question (a) — CAIE IGCSE Mathematics Extended

Scale: 5 cm to 200 m

[ 1 ]

(i)

Measure angle ABC.

Angle A B C=

[ 1 ]

(b)

A bronze statue is 4.5 m high and has a mass of 195200 kg .

The density of bronze is $8000 \mathrm{~kg} / \mathrm{m}^{3}$ .
The volume of a mathematically similar model of the statue is $0.385 \mathrm{~m}^{3}$ .
Calculate the height of the model.
[Density = Mass $\div$ Volume ]

[ 5 ]

1.13 or 1.128 to 1.129
M4 for $4.5\sqrt[3]{\frac{0.385\times8000}{195200}}$ oe, or $\sqrt[3]{\frac{4.5^3\times0.385\times8000}{195200}}$ oe
or M3 for $\sqrt[3]{\frac{0.385}{\text{their }24.4}}$ or $\sqrt[3]{\frac{\text{their }3080}{195200}}$ , or $\frac{0.385}{\text{their }24.4}=\frac{l^3}{4.5^3}$ oe
or M2 for $\frac{\text{their }24.4}{0.385}$ or $\frac{0.385}{\text{their }24.4}$ oe, or B2 for 24.4 or 3080 seen
or M1 for $195200\div8000$ or for $0.385\times8000$

[Maximum number: 3]

A lake has an area of $3 \mathrm{~km}^{2}$ .
On a map the area of the lake is $18.75 \mathrm{~cm}^{2}$ .
Find the scale of the map in the form 1: n.

40000

B2 for 1 cm to 0.4 km or 2.5 cm to 1 km or 1600000000
or M2 for $\sqrt{\frac{3\times10^k}{18.75}}$ oe where k>5
or M1 for area conversion evidence listed in markscheme

[Maximum number: 5]

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.

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: 7]

(a)

The diagram shows three lines meeting at a point.

The ratio a: b: c=3: 4: 5.

Find the value of c.

NOT TO SCALE

[ 3 ]

150
M1 for $\frac{k}{3+4+5}\times p$ , where p=1,3,4 or 5, or $\frac5{12}$ oe
B1 for 360 used

(b)

A regular pentagon has an exterior angle, d.

A regular hexagon has an interior angle, h.
Find the fraction $\frac{d}{h}$ .
Give your answer in its simplest form.

[ 4 ]

$\frac35$ cao nfww
B3 for $\frac{72}{120}$ , or B2 for d=72 or h=120, or M1 for $360\div5$ oe or $180-(360\div6)$ oe

[Maximum number: 2]

NOT TO SCALE

A, B and C are points on the circle, centre O.
DE is a tangent to the circle at C.
AC=10 cm, AB=9.5 cm and BC=7.7 cm.

(a)

Find

[ 2 ]

(i)

angle AOC

[ 1 ]

(ii)

angle ACD.

\text { Angle } A C D=

[ 1 ]

70.2
FT $90-\text{their (b)(ii)}$

[Maximum number: 5]

(a)

The interior angle of a regular polygon with n sides is $150^{\circ}$ .
Calculate the value of n.

[ 2 ]

12
M1 for $150=\frac{(n-2)\times180}{n}$ or $\frac{360}{180-150}$ oe.

(b)

Figure for Question (b) — CAIE IGCSE Mathematics Extended

NOT TO SCALE
A, B, C and D are points on the circle, centre O.
M is the midpoint of AB and N is the midpoint of CD.
O M=O N

Explain, giving reasons, why triangle OAB is congruent to triangle OCD.

[ 3 ]

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Covered syllabus points

4. Geometry question 1

Question (a)

Question (i)

Question (b)

4. Geometry question 2

4. Geometry question 3

Question (a)

Question (b)

Question (c)

4. Geometry question 4

Question (a)

Question (b)

4. Geometry question 5

Question (a)

Question (i)

Question (ii)

4. Geometry question 6

Question (a)

Question (b)