Geometric Constructions
Geometry · Geometric Constructions
Syllabus tag: KCSE | Mathematics | Form 1 | Topic 21 Geometric Constructions
Lesson objectives
By the end of this topic, you should be able to:
- Construct a perpendicular bisector of a line.
- Construct an angle bisector.
- Construct a perpendicular to a line from a given point.
- Construct angles of 90°, 60°, 45°, 30° and their combinations, and construct triangles.
Geometric Constructions
A construction uses only a ruler and a pair of compasses. No protractor, and no measured angles.
Leave all construction arcs visible. They are the evidence of method, and marks are given for them.
a) The perpendicular bisector
Open the compasses to more than half the line.
Draw arcs above and below from each end, keeping the same radius.
Join the two crossing points.
The result is perpendicular to the line and passes through its midpoint.
If the radius is less than half the line, the arcs never meet.
Every point on this bisector is equidistant from the two ends of the original line. That property is what makes it useful for finding circle centres.
b) The angle bisector
Place the compass point at the vertex and draw an arc cutting both arms.
From each of those two points, draw equal arcs that cross.
Join the vertex to the crossing point.
Every point on the bisector is equidistant from the two arms.
c) Perpendicular from a point on a line
Place the compass point at the given point and draw an arc cutting the line on both sides.
Then construct the perpendicular bisector of the segment between those two cuts.
It passes through your chosen point at 90°.
d) Perpendicular from a point off the line
Place the compass point at the external point and draw an arc cutting the line twice.
Bisect the segment between the two cuts.
The bisector passes through the external point and meets the line at a right angle. This is the shortest distance from the point to the line.
e) Constructing standard angles
90° comes from a perpendicular.
60° comes from an equilateral triangle. Draw a line, then an arc from a point on it. From where that arc cuts the line, draw a second arc of the same radius. Join to the crossing.
45° is 90° bisected. 30° is 60° bisected. 15° is 30° bisected.
Angles combine. Place 60° next to 15° for 75°. Place 90° next to 30° for 120°.
f) Constructing triangles
Given three sides: draw one as the base, then arc the other two lengths from each end. Where the arcs cross is the third vertex.
Given two sides and the included angle: construct the angle first. Mark the two lengths along its arms, then join.
Given two angles and a side: draw the side, then construct one angle at each end. Extend the arms until they meet.
g) Where this is used
Surveying without instruments. Setting out foundations. Technical drawing. Any figure that must be exact rather than measured by eye.
Words to know
- Construction -- drawing an exact figure using only ruler and compasses.
- Perpendicular bisector -- a line cutting another in half at right angles.
- Angle bisector -- a line dividing an angle into two equal parts.
- Equidistant -- at an equal distance from two or more points.
- Locus -- the set of all points satisfying a given condition.
:::checkpoint Check yourself
- Why must the compass radius exceed half the line when bisecting it?
- How do you construct an angle of 30°?
- What is special about every point on a perpendicular bisector?
- Which construction gives the shortest distance from a point to a line? :::
Bridge to practice
The exercises begin with what construction permits, move through the bisectors and perpendiculars, and finish with standard angles and triangles. When practising, use a sharp pencil and check each finished construction with a protractor — not to produce it, but to confirm your accuracy.