IGCSE Maths: Geometrical Constructions

Name: ................................................................................... Group: .........................................

Introduction

A geometrical construction is an accurate geometric drawing. They are usually constructed with
a ruler (sometimes called a straight-edge) and a pair of compasses. Other instruments may be
used, such as a protractor and set-squares.

Pair of compasses Protractor Set-square (30°-60°-90°)

With these instruments, we can draw and mark lengths, draw circles, construct and bisect angles
and lines.

I) Constructing a triangle given the lengths of sides

Worked Example: Construct triangle ABC with AB=8.5 cm, BC=7.2 cm and CA=6.9 cm.

C
6.9 7.2

A 8.5 B A B A B
8.5 cm 8.5 cm

(i) sketch (ii) draw side AB (iii) mark C from A

C C

6.9 cm 7.2 cm

A B A B A B
8.5 cm 8.5 cm 8.5 cm

(iv) mark C from B (v) intesect to find C (vi) draw sides and label

Once you have finished your construction, it is not necessary to remove any construction lines.
These will help your examiner see how you made the construction.
II) Constructing a triangle with a given angle

Worked Example: Construct triangle PQR with PQ=4 cm, QR=6.5 cm and ∠PQR=60°.

P
4
60°
60°
Q 6.5 R Q R
6.5 cm Q 6.5 cm R

(i) sketch (ii) draw side QR (iii) draw the 60° line

P P P

4 cm 4 cm 4 cm
60° 60° 60°
Q 6.5 cm R Q 6.5 cm R Q R
6.5 cm

(iv) measure and mark P (v) connect PR (vi) finished

III) Constructing a perpendicular bisector of a line-segment

Worked Example: Construct the perpendicular bisector AB of segment XY

90°
X Y
X Y

B X Y

(i) sketch (ii) draw segment XY (iii) draw a radius from X

A
A

90°

X Y X Y X Y

B B

(iv) draw an equal radius from Y (v) draw AB (vi) measure lengths and angles

The perpendicular bisector is a very useful construction; we will use it in two more
constructions that follow, to find the circumcentre of a triangle and the centre of a circle. The
circumcentre of a triangle is the centre of a circle with a radius that intersects all the vertices of
the triangle. This point can be easily found using a perpendicular bisector. Similarly the centre of
a circle may be found by bisecting two given chords.
IV) Constructing the circumcentre of a triangle

Worked Example: Find the circumcentre P of triangle ABC

A A

B C
B C

B C

(i) sketch (ii) draw triangle ABC (iii) construct the bisector of AB
A

A A

B C
P

B C B C

(iv) construct the bisector of AC (v) repeat for BC and mark centre P (vi) draw the circle centred on P

V) Finding the centre of a given circle

(i) sketch (ii) draw 2 chords on a circle (iii) bisect the first chord

(iv) bisect the second chord (v) intersect the bisectors (vi) mark the centre

The above method will work for any circle provided that the two chords drawn on the circle are
not parallel to one another.

Douglas C Colgan 31/10/2020.