Geometrical Constructions
4.0 Geometry · 4.1 Geometrical Constructions
Syllabus tag: Kenya CBC | Grade 8 Mathematics | Strand 4.0 Geometry | Sub-Strand 4.1 Geometrical Constructions (12 lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Construct parallel and perpendicular lines, and divide a line proportionally
- Identify angle properties of polygons
- Construct regular and irregular polygons up to a hexagon
- Construct circles passing through the vertices of a triangle, and circles touching the sides of a triangle
Geometrical Constructions
A construction is a drawing made with only a ruler and a pair of compasses. No protractor, no measuring of angles. Everything is built from equal distances.
Work with a sharp pencil and leave your construction arcs on the page. They are the evidence that the drawing was constructed rather than guessed.
a) The perpendicular bisector
The perpendicular bisector of a line cuts it exactly in half at right angles.
Open the compasses to more than half the line. Draw an arc from each end, above and below. Join the two crossing points.
The radius must exceed half the line, or the arcs never meet. Keep the same radius for both ends, since that equal distance is what makes the construction work.
Every point on this line is the same distance from both ends of the original line.
b) 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.
Join the vertex to the crossing point. That line splits the angle into two equal halves.
c) Perpendicular and parallel lines
To drop a perpendicular from a point to a line, draw an arc from the point. It should cut the line twice. Then bisect the segment between those two cuts.
For a parallel line, construct two perpendiculars to the same line at different places. Mark equal distances along them. Joining those marks gives the parallel.
d) Dividing a line proportionally
Say you must divide a line in the ratio 2 : 3. Draw a second line from one end, at any convenient angle.
Step off 5 equal lengths along it with the compasses, since 2 + 3 is 5.
Join the last mark to the far end of the original line. Then draw lines parallel to that one through the other marks. They cut the line in the ratio asked.
e) Angles in polygons
The exterior angles of any polygon add to 360°, however many sides it has.
An interior angle and its exterior angle sit on a straight line, so together they make 180°.
For a regular polygon with n sides, one exterior angle is 360 ÷ n. The interior angle is 180 minus that.
f) Constructing polygons
A regular hexagon is the easiest. Draw a circle, then step the radius round the circumference six times. The six marks are the vertices.
That works because the side of a regular hexagon equals the radius of its circle.
For other regular polygons, work out the angle at the centre. It is 360 ÷ n. Mark that angle round from the centre.
g) Circles and triangles
The circumcircle passes through all three vertices. Construct the perpendicular bisectors of two sides. Where they cross is the centre, and the distance to any vertex is the radius.
The incircle touches all three sides. Construct the bisectors of two angles. Where they cross is the centre, and the perpendicular distance to any side is the radius.
Notice which construction each one uses. Bisecting sides finds a point equally far from the vertices. Bisecting angles finds a point equally far from the sides.
h) Where this is used
A carpenter finds the centre of a board. A surveyor sets out a right angle without instruments. A designer constructs a regular shape that has to be exact.
Words to know
- Parallel lines -- lines that stay the same distance apart and never meet.
- Perpendicular lines -- lines that meet at a right angle (90°).
- Regular polygon -- a polygon with all sides and all angles equal.
- Circumscribed circle -- a circle passing through all the vertices of a polygon (such as a triangle).
- Inscribed circle -- a circle touching all the sides of a polygon (such as a triangle) without crossing them.
:::checkpoint Check yourself
- Why must the compass radius be more than half the line when bisecting it?
- What do the exterior angles of any polygon add up to?
- Find the interior angle of a regular octagon.
- Which construction finds the centre of a circle through all three vertices? :::
Bridge to practice
Try the exercises below -- on parallel and perpendicular lines, polygon angle properties, and circles related to triangles.