Edexcel IGCSE Math B 6 Geometry Questions

Practise angle facts, congruence, similarity, loci, symmetry, Pythagoras, and circle geometry using diagrams and written reasons.

Covered syllabus points

6.A Euclidean geometry 6.B Geometrical reasoning 6.C Angle properties 6.D Quadrilaterals 6.E Symmetry 6.F Pythagoras in 2D and 3D 6.G Similarity 6.H Triangle similarity 6.I Congruent shapes 6.J Triangle congruence 6.K Circle theorems 6.L Cyclic quadrilaterals 6.M Loci 6.N Constructions

[Maximum number: 2]

Each interior angle of a regular polygon is $172^{\circ}$

Calculate the number of sides of the polygon.

The length of each side of the regular polygon is 5.2 cm to 2 significant figures.

$\frac{360}{180-172}=45$
Final answer:
45

Marking notes:
Equivalent equation (n-2)180=172n is allowed.

[Maximum number: 3]

Figure for Question 2 — Edexcel IGCSE Math B

Diagram NOT accurately drawn.

Figure 2 shows a quadrilateral ABCD with
$\overrightarrow{AB}=\binom{8.5}{4.5}\quad\text{and}\quad \overrightarrow{AC}=\binom{15}{5}.$
The diagonals AC and BD meet at the point E.

Given that ABCD is a kite, find the coordinates of the point D.

$\overrightarrow{OB}=\binom{-3}{1}+\binom{8.5}{4.5}=\binom{5.5}{5.5}$ $\overrightarrow{BE}=\binom{-8.5}{-4.5}+\binom{3}{-1}+\binom{6}{4}=\binom{0.5}{-1.5}$ $\overrightarrow{OD}=\overrightarrow{OE}+\overrightarrow{BE} =\binom{6}{4}+\binom{0.5}{-1.5} =\binom{6.5}{2.5}$

Final answer:
(6.5,2.5)

[Maximum number: 2]

Figure for Question 3 — Edexcel IGCSE Math B

Write down

(a)

the number of lines of symmetry of the figure,

[ 1 ]

(b)

the order of rotational symmetry of the figure.

[ 1 ]

[Maximum number: 2]

Figure for Question 4 — Edexcel IGCSE Math B

Figure 3 shows a solid right prism ABCDEFGH.
Diagram NOT accurately drawn.
The base of the prism, ABCD, is a rectangle and is horizontal. Trapezium BCHG is a cross section of the prism.
$AB=4\text{ m},\quad BC=3\text{ m},\quad CH=2.4\text{ m},\quad GB=1.5\text{ m}$ $\angle GBC=\angle BCH=\angle ABG=\angle DCH=90^\circ$

Calculate the length, in metres to 3 significant figures, of AH.

$AH=\sqrt{4^{2}+3^{2}+2.4^{2}} = \sqrt{30.76}$
Final answer:
$5.55\text{ m}$

[Maximum number: 3]

Figure 1 shows triangle OAB in which $\overrightarrow{O A}=4 \mathbf{a}$ and $\overrightarrow{O B}=8 \mathbf{b}$
P is the point on OA such that O P: O A=1: 4

The area of APQB is 150 cm².

Calculate the area of triangle OPQ.

Triangles OAB and OPQ are similar.
$|\triangle OAB|=4^{2}|\triangle OPQ|$

The area of APQB is:
$150=|\triangle OAB|-|\triangle OPQ|$ $150=4^{2}|\triangle OPQ|-|\triangle OPQ|$ $150=15|\triangle OPQ|$ $|\triangle OPQ|=10$

Final answer:
$10\text{ cm}^{2}$

M1: similar-triangle area scale factor.
M1 (dep): using the difference of the two triangle areas for APQB.
A1.

[Maximum number: 3]

Figure 2

Diagram NOT accurately drawn

Figure 2 shows a kite ABCD in which

A B=6 \mathrm{~cm} \quad B C=8 \mathrm{~cm} \quad \angle A B C=90^{\circ}

AEC and BED are straight lines.

Prove that triangle ABE is similar to triangle BCE.

[Maximum number: 3]

Figure for Question 7 — Edexcel IGCSE Math B

The diagram shows a square ABCD.
M is the midpoint of AD.
N is the midpoint of AB.
Prove that $\triangle DAN$ is congruent to $\triangle CDM$ .

Valid congruence statements include:

AD=DC
because ABCD is a square.

AN=DM
because N and M are midpoints of AB and AD.

$\angle DAN=\angle CDM=90^\circ$
because the angles are right angles in the square.

Therefore $\triangle DAN\cong\triangle CDM$ by SAS.

M2 for two correct statements with reasons.
M1 for one correct statement with reason, or all three statements with no reasons.
A1 for SAS/RHS/SSS stated with all three correct reasons.

[Maximum number: 3]

Diagram NOT accurately drawn

Figure 3 shows a circle with centre O and radius 10 cm.
The points A and B lie on the circle such that $\angle A O B=100^{\circ}$ The point C is such that AC and BC are tangents to the circle.

Giving reasons, calculate the size, in degrees, of $\angle A C B$ .

$\angle OAC=\angle OBC=90^\circ$ $\angle ACB=80^\circ$
Reasons may use tangent-radius angles and angles in a quadrilateral, or split into two triangles using tangent, triangle angle sum, and the isosceles/bisector facts.

[Maximum number: 2]

Diagram NOT accurately drawn

In the diagram, A, B, C and D are points on a circle, centre O. $\angle C B O=62^{\circ}$ and $\angle B C D=126^{\circ}$

Find the size, in degrees, of $\angle B A D$ .

Give a reason for your answer.

\angle B A D=\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .

CFDE is cyclic, so opposite angles sum to $180^\circ$ :
$\angle CDE=180^\circ-126^\circ=54^\circ.$

Answer: $54^\circ$ .

Marks: B1.

[Maximum number: 2]

The scale drawing opposite shows the positions, A, B and C, of three lighthouses. The points A, B and C lie at sea level in the same horizontal plane with B due north of A.

The scale of the drawing is 1 cm represents 50 km.

A ship S is equidistant from A and C and 225 km from B.

Showing all your construction lines, find and mark each of the two possible positions of S on the scale drawing with a cross (×).

[Maximum number: 4]

Figure for Question 11 — Edexcel IGCSE Math B

ABCD is a quadrilateral.
Leaving in all of your construction lines, construct the locus of all points inside the quadrilateral that are

(a)

equidistant from B and C,

[ 2 ]

(b)

equidistant from the lines DA and DC.

[ 2 ]

Correct angle bisector of $\angle ADC$ .

Marks: M1 for construction arcs from D and from points on DA, DC, A1 dependent for correct angle bisector.

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Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question (a)

Question (b)