Loci and Constructions: KS3 Maths Guide

A locus (plural: loci) is the set of all points that satisfy a given rule. Constructions use only a ruler and compasses — no protractor, no measuring — to draw exact geometric shapes. Together, loci and constructions form a core KS3 geometry topic that rewards careful, patient technique.

What equipment do you need for constructions?

At KS3, all constructions must be done with a sharp pencil, a ruler (for straight edges only — not for measuring lengths), and a pair of compasses. A rubber is useful for faint construction arcs that you no longer need, but keep all arcs visible during the working so your examiner can see your method.

Never use a protractor to construct an angle bisector or perpendicular bisector — the whole point is to produce an exact result using arcs and intersections alone.

How do you construct the perpendicular bisector of a line segment?

The perpendicular bisector of a line AB passes through its midpoint at 90°. Every point on the perpendicular bisector is equidistant from A and B.

Step 1 — Open your compasses to more than half the length of AB (a setting between ½ and ¾ works well).
Step 2 — Place the compass point on A and draw an arc above and below the line.
Step 3 — Without changing the compass width, place the point on B and draw two more arcs (above and below) that cross the first pair.
Step 4 — Use a ruler to draw a straight line through both intersection points.

The line you have drawn is the perpendicular bisector. It crosses AB at its exact midpoint and makes a 90° angle.

How do you construct the angle bisector?

The angle bisector divides any angle exactly in half. It is also the locus of points equidistant from the two arms of the angle.

Step 1 — Place the compass point on the vertex of the angle. Draw an arc that crosses both arms of the angle.
Step 2 — Place the compass point on each intersection in turn (keep the same compass width) and draw two arcs inside the angle that cross each other.
Step 3 — Draw a straight line from the vertex through the intersection of the two inner arcs.

This bisector line cuts the original angle exactly in two.

What are the main loci to know at KS3?

A locus describes where a point can be, given a distance condition. The four classic loci are:

Condition | Locus produced
Fixed distance r from a point P | Circle, centre P, radius r
Equal distance from two points A and B | Perpendicular bisector of AB
Fixed distance d from a straight line | Two parallel lines, each distance d from the original
Equal distance from two lines meeting at a point | Angle bisector of those two lines

Recognising which condition applies is half the battle in any locus question.

How do you construct an equilateral triangle?

An equilateral triangle has three sides of equal length and three 60° angles. Here is how to construct one with side length 5 cm.

Step 1 — Draw a base line AB of length 5 cm.
Step 2 — Set the compasses to 5 cm. Place the point on A and draw an arc above the line.
Step 3 — Keep the compasses at 5 cm. Place the point on B and draw another arc above the line, crossing the first.
Step 4 — The intersection of the two arcs is point C. Join A to C and B to C.

All three sides are exactly 5 cm and all angles are exactly 60°.

How do you draw a region satisfying more than one condition?

Examination questions often ask you to shade a region that satisfies two or more locus conditions simultaneously. For example: shade the region that is less than 4 cm from A AND closer to B than to A.

Step 1 — Draw the circle of radius 4 cm centred on A (the boundary for the first condition).
Step 2 — Draw the perpendicular bisector of AB (the boundary for the second condition).
Step 3 — The required region is inside the circle AND on the B-side of the perpendicular bisector.
Step 4 — Shade the overlapping area carefully and label it.

Always draw the boundary lines first, then identify and shade the correct region.

What mistakes do students commonly make?

Rubbing out construction arcs. Keep all arcs. Without them, your working is invisible and you lose method marks.

Not keeping the compass width constant between arcs in the same construction. Even a tiny change makes the intersection points incorrect.

Using a protractor to bisect angles or draw perpendicular lines — this does not count as a construction.

Confusing the locus of a point at distance d from a line — many students draw only one parallel line instead of two (one on each side).

Frequently asked questions

What is the difference between a construction and a drawing?

A drawing uses measurements — a ruler marked in centimetres, a protractor, and so on. A construction uses only a ruler as a straight edge and compasses to produce a geometrically exact result. In exam mark schemes, "construct" means no protractor may be used; "draw" means ordinary measuring instruments are allowed.

Do I have to leave in the construction arcs?

Yes, always. The arcs are your method. An examiner cannot award method marks if they cannot see the arcs, even if your final line is correct. Only rub out arcs when a question specifically tells you to.

What is the locus of a point that moves so it is always 3 cm from a fixed point?

It is a circle with radius 3 cm centred on the fixed point. Every point on that circle is exactly 3 cm from the centre, so the circle is the complete set of all such points.

Can loci questions appear with regions?

Yes, and this is very common at KS3 and GCSE. A question might ask you to shade the region satisfying two or three conditions at once — for example, within 5 cm of a point AND closer to one wall than to another. Sketch each boundary first, then shade the overlap.

For Socratic KS3 geometry practice, see aitutors.me.