CAIE IGCSE Mathematics Extended 4 Geometry

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: 7]

(a)

Figure for Question (a) — CAIE IGCSE Mathematics Extended

NOT TO
SCALE
ABCDEF is a regular hexagon.
D F, D A and DB are diagonals.
Complete the following statements using three different triangles.
Triangle DEF is congruent to triangle ......
Triangle ...... is congruent to triangle ......

[ 2 ]

(b)

Figure for Question (b) — CAIE IGCSE Mathematics Extended

NOT TO SCALE
P and Q are points on the circle with centre O.
TP and TQ are tangents to the circle from the point T.
Complete the following statements and reasons.
In triangles OPT and OQT
O P= ...... because each is a radius of the circle
OT is a common side

Angle O P T= angle ...... =90^ because ......
Triangles OPT and OQT are congruent using the criterion ......
This proves that the tangents TP and TQ are ......

[ 5 ]

[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: 15]

(a)

The diagram shows an isosceles triangle with the base extended.
Find the value of x.

[ 3 ]

111
M2 for $180-\frac{180-42}{2}$ oe or $42+\frac{180-42}{2}$ oe, or M1 for $\frac{180-42}{2}$ oe

(b)

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

(c)

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

(d)

Figure for Question (d) — CAIE IGCSE Mathematics Extended

Show that PQRS is a cyclic quadrilateral.

[ 5 ]

x+2 x-5+x+20+3 x-40=360
Accept equivalent equation

e.g. 7 x-25=360

7 x=360+5-20+40 or better
FT their equation, accept e.g. 7 x=385

x=55
55 and 125
or 105 and 75

B1dep

Dep on M1M1B1

Accept $55+3 \times 55-40=180$

or $2 \times 55-5+55+20=180$

If B0 scored, SC1 for 55, 75, 105 and 125

Opposite angles sum to 180 oe
[so PQRS is a cyclic quadrilateral ]
Dep on M1M1B1B1

[Maximum number: 5]

NOT TO SCALE

The diagram shows a cyclic quadrilateral ABCD. BD and AC intersect at X.

(a)

Angle $B A D=74^{\circ}$ and angle $B C A=34^{\circ}$ .

Find

[ 3 ]

(i)

angle BDA

Angle B D A=

[ 1 ]

(ii)

angle BCD

Angle B C D=

[ 1 ]

(iii)

angle ABD.

Angle A B D=

[ 1 ]

(b)

In the diagram, triangle ADX is similar to triangle BCX.
$B C=4.5 \mathrm{~cm}, A D=9 \mathrm{~cm}$ and $C X=3.3 \mathrm{~cm}$ .

Work out XD.
X D= cm [2]

[ 2 ]

6.6

M1 for $\frac{4.5}{9}=\frac{3.3}{XD}$ oe.

All question bank results loaded

CAIE IGCSE Mathematics Extended 4 Geometry Topic Practice

Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question 3

Question (a)

Question (b)

Question (c)

Question (d)

Question 4

Question (a)

Question (i)

Question (ii)

Question (iii)

Question (b)