Ruler-and-compass constructions

The four standard constructions — perpendicular bisector, angle bisector, perpendicular from/at a point, and 60° — why each one works, and why the construction arcs must be left on the page.

Geometry & measures · Constructions & loci

Ruler-and-compass constructions

The four standard constructions — perpendicular bisector, angle bisector, perpendicular from/at a point, and 60° — why each one works, and why the construction arcs must be left on the page.

Why it works

A pair of compasses draws "all the points this far from here". Every standard construction is built from that one idea, so each one can be reasoned rather than memorised.

Perpendicular bisector of $AB$ . Open the compasses to more than half of

AB

. With the point on

A

, draw arcs above and below; keeping the *same radius*, repeat from

B

. Each crossing point is the same distance from

A

as from

B

— and the set of points equidistant from

A

and

B

is exactly the perpendicular bisector. Join the two crossings.

Angle bisector. Put the compass point on the vertex and draw an arc crossing both arms — that marks two points the same distance along each arm. From those two marks, draw equal arcs that cross inside the angle. That crossing is equally distant from both arms, so joining it to the vertex cuts the angle exactly in half.

Perpendicular from a point to a line (dropping a perpendicular): from the point, draw an arc cutting the line twice, then perpendicular-bisect that pair of crossings. Perpendicular at a point on a line: mark equal distances either side of the point along the line first, then bisect.

Constructing 60°. An equilateral triangle has three

60°

angles, so: draw an arc from one end of a line, then — same radius — an arc from where it crosses. Joining up gives a triangle with all sides equal, so the angle is

60°

. Bisect it for

30°

; bisect a right angle for

45°

.

Two rules that lose more marks than the geometry.

Keep the same compass radius wherever the method says so. Adjusting

the compasses part-way makes the "equal distances" argument false, and the construction is wrong even if the picture looks right.

Leave all construction arcs visible. The arcs are the evidence, and

the mark scheme awards them. A perfect line measured with a protractor and ruler — arcs rubbed out — scores nothing.

Use a sharp pencil and construct, never measure. If a question says "construct", a protractor is not a valid tool for the angle.