17.3 Constructions with ruler and compasses — Learn

Prerequisites: measure a line segment and recognise a right angle.

Objectives: construct a perpendicular bisector, a perpendicular through or from a point, an angle bisector and a $60^\circ$ angle. This page covers G2.

Perpendicular bisector

A perpendicular bisector meets a segment at $90^\circ$ and cuts it into two equal parts. Keep the compass width greater than half the segment.

Worked construction. Bisect segment $AB$.

1. Open the compasses to more than half of $AB$.
2. With centre $A$, draw arcs above and below the segment.
3. Keeping the same width, repeat from centre $B$.
4. Draw a straight line through the two arc intersections.

The intersections are equally far from $A$ and $B$, so their joining line is the perpendicular bisector.

Perpendiculars and angle bisectors

To construct a perpendicular to a line at a point on it, mark two points on the line at equal distance from the point. Draw equal-radius arcs from those two marks, then join their intersection to the point. For a point away from the line, use arcs centred at the point to mark two equal-distance points on the line, then construct the perpendicular bisector between them.

To bisect an angle, draw an arc from its vertex to meet both arms. From those two marks, draw equal arcs that cross inside the angle. Join the vertex to the crossing point.

Worked construction. Construct $60^\circ$ at point $A$ on a ray.

1. Draw an arc centred at $A$ to meet the ray at $B$.
2. With the same compass width and centre $B$, draw an arc meeting the first arc at $C$.
3. Draw ray $AC$.
4. Since $AB=BC=CA$, triangle $ABC$ is equilateral and each angle is $60^\circ$.

Watch out. Changing the compass width between paired arcs breaks the equal-distance argument. Keep one setting until the intersections are marked.

Guided investigation: where do the arcs meet?

Predict: will the two arc intersections be the same distance from $A$ and $B$? Write your prediction down first.

Test: follow the displayed construction and compare those distances.

Explain: why does joining the intersections make a perpendicular bisector? Write your explanation before reading on.

Progress check

The garden centre is halfway across the garden in both directions. Describe a construction that finds the centre, and explain why the two bisectors meet there.

You should now be able to

• construct a perpendicular bisector with equal-radius arcs;
• construct a perpendicular at or from a point;
• bisect an angle and construct $60^\circ$;
• explain why the compass arcs establish equal distances.