Geometrical Constructions
4.0 Geometry · 4.2 Geometrical Constructions
Syllabus tag: Kenya CBC | Grade 7 Mathematics | Strand 4.0 Geometry | Sub-Strand 4.2 Geometrical Constructions (12 lessons)
Lesson objectives
By the end of this topic, you will be able to:
- Bisect angles using a ruler and a pair of compasses only
- Construct 90°, 45°, 60°, 30° and other angles that are multiples of 7.5°, using a ruler and a pair of compasses only
- Construct triangles and circles using a ruler and a pair of compasses only
Geometrical Constructions
A construction uses only a ruler and a pair of compasses. No protractor, and no measuring of angles.
Leave your arcs on the page. They show the work was constructed rather than guessed.
a) Bisecting an angle
Put the compass point on the vertex and draw an arc cutting both arms.
From each of those two points, draw equal arcs that cross each other.
Join the vertex to the crossing point. That line cuts the angle exactly in half.
Keep the same compass opening for the two crossing arcs. That equal distance is what makes the construction work.
b) The perpendicular bisector
Open the compasses to more than half the line. Draw an arc from each end, above and below. Join the two crossing points.
The line produced is perpendicular to the original and passes through its midpoint.
If the radius is less than half the line, the arcs never meet. There is then nothing to join.
c) Constructing 90°
A perpendicular gives a right angle directly.
Draw a line, mark a point on it, and construct the perpendicular there. The angle between them is 90°.
d) Constructing 60°
An equilateral triangle has three 60° angles, and it can be built with compasses alone.
Draw a line. From one end, draw an arc of any radius. From where that arc meets the line, draw another arc of the same radius. Join the starting point to where the arcs cross.
That angle is exactly 60°, because all three sides of the triangle are equal.
e) Building the other angles
Every other angle in the syllabus comes from halving.
Bisect 90° to get 45°. Bisect 45° to get 22.5°.
Bisect 60° to get 30°. Bisect 30° to get 15°. Bisect 15° to get 7.5°.
Angles can also be added. Place a 60° next to a 30° and you have 90°. Place 60° and 15° together for 75°.
f) Constructing triangles
Given three sides, draw one side as a base. Set the compasses to the second side and draw an arc from one end. Set them to the third side and draw an arc from the other end. Where the arcs cross is the third vertex.
Given two sides and the angle between them, construct the angle first. Mark the two side lengths along its arms, then join.
g) Constructing circles
A circle needs a centre and a radius. Set the compasses to the radius, place the point at the centre, and turn once.
To draw a circle through three points, construct the perpendicular bisectors of two of the joining lines. Where they cross is the centre.
h) Where this is used
A carpenter marks a right angle without a set square. A surveyor divides a plot. A designer constructs an exact shape. Anyone drawing a plan to scale relies on these methods.
Words to know
- Geometrical construction — drawing accurate shapes and angles using only a ruler and a pair of compasses.
- Bisect — to divide something (an angle or a line) exactly into two equal halves.
- Equilateral triangle — a triangle with all three sides equal and all three angles equal to 60°.
:::checkpoint Check yourself
- What two instruments may be used in a construction?
- Why must the compass radius exceed half the line when bisecting it?
- How do you construct a 30° angle?
- Which construction finds the centre of a circle through three points? :::
Bridge to practice
Try the exercises below — on bisecting angles, constructing common angles, triangles and circles.