Constructions, Loci and Scale Drawings — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the standard ruler-and-compass constructions, the idea of a locus, and the use of scale drawings to answer practical questions. By the end of this guide you should be able to construct a perpendicular bisector, an angle bisector, a perpendicular from or to a line, and a 60° angle.

You should also be able to describe and draw the four basic loci, shade a region satisfying several conditions at once, and combine constructions with a scale to solve a real problem.

The organising idea is that every construction is really a statement about equal distances. A perpendicular bisector is the set of points equidistant from two points; an angle bisector is the set of points equidistant from two lines; a circle is the set of points a fixed distance from one point. That is why the compasses do all the work and the protractor does none — the compasses are the tool that guarantees two distances are equal. Understanding which equal-distance condition a construction produces also tells you immediately which locus it draws, so the two halves of the topic are the same idea twice.

Key terms and definitions

Construction — an accurate drawing made with a ruler and compasses only.

Construction arcs — the compass marks left on the page. They must not be rubbed out; they are the evidence of the method.

Perpendicular bisector — a line at right angles to a segment, cutting it exactly in half.

Angle bisector — a line cutting an angle exactly in half.

Locus (plural loci) — the set of all points satisfying a given condition.

Equidistant — the same distance from two or more things.

Region — the area satisfying a condition, usually shaded on the diagram.

Core concepts

The one rule about equipment

Constructions are done with a ruler and compasses only. A protractor is not acceptable for a construction, even if the result looks right, because the question is testing the method rather than the drawing.

Leave all construction arcs visible. Marks are awarded for them, and an accurate line with no arcs typically scores nothing. Erasing them is one of the most common self-inflicted losses in the topic.

Use a sharp pencil, and keep the compasses at a fixed radius while drawing a pair of arcs that are meant to be equal.

Perpendicular bisector of a line segment

Open the compasses to more than half the length of the segment. Place the point at one end and draw arcs above and below the line; repeat from the other end with the same radius. Join the two crossing points.

The result is perpendicular to the segment and cuts it exactly in half.

Every point on this line is equidistant from the two ends of the segment, which is what makes it a locus as well as a construction.

If the compasses are opened to less than half the length, the arcs never meet — which is the usual reason a construction fails to work.

Angle bisector

Place the compass point on the vertex and draw an arc crossing both arms of the angle. From each of those two crossings, draw arcs of equal radius that meet each other. Join the vertex to that meeting point.

Every point on the bisector is equidistant from the two arms.

The same construction is used whatever the size of the angle, and it works for reflex angles too.

Perpendicular from a point to a line

To drop a perpendicular from a point down to a line, place the compass point on the point and draw an arc crossing the line twice. Then construct the perpendicular bisector of the segment between those two crossings.

The resulting line passes through the original point and meets the line at right angles.

The shortest distance from a point to a line is measured along this perpendicular, which matters in locus questions about distance from a wall or a road.

Perpendicular at a point on a line

To erect a perpendicular at a point that lies on the line, draw two arcs of equal radius from that point, one each side, to mark two points on the line. Then construct the perpendicular bisector of those two marks.

Constructing a 60° angle

An equilateral triangle has three 60° angles, so constructing one gives a 60° angle without a protractor.

Draw a line, place the compass point at one end and draw an arc; keeping the same radius, draw an arc from where the first arc crossed the line. Join the original point to where the arcs meet.

Bisecting that angle gives 30°, and bisecting a right angle gives 45° — so several standard angles are available by construction alone.

The four basic loci

A fixed distance from a point gives a circle of that radius.

A fixed distance from a line segment gives a "racetrack" shape: two parallel lines either side, joined by semicircular ends.

Equidistant from two points gives the perpendicular bisector of the segment joining them.

Equidistant from two lines gives the angle bisector of the angle between them.

The last two are exactly the constructions above, which is why the constructions must be learned properly before locus questions become straightforward.

Regions and inequalities

Many questions ask for a region rather than a line — "less than 4 cm from A", or "nearer to B than to C".

Draw the boundary first using the appropriate construction, then decide which side satisfies the condition and shade it.

"Nearer to B than to C" means the side of the perpendicular bisector containing B. Testing a single obvious point is the quickest way to confirm which side is which.

Where several conditions apply at once, draw all the boundaries, then shade only the region satisfying every one of them. Labelling the region clearly, and stating what the shading means, is expected.

A boundary is included when the condition says "at most" or "at least", and excluded when it says "less than" or "more than" — some mark schemes ask for a dashed line in the second case.

Scale drawings

A scale drawing turns a real situation into an accurate diagram, so lengths can be measured rather than calculated.

Convert real distances using the scale, construct the diagram accurately, then measure the answer and convert back.

Typical questions combine a scale with bearings or with a locus — finding where a ship can be, or where a mast must go to be within range of two towns.

Work to about one millimetre and one degree, and always state the scale you used.

Worked examples

Example 1: Equidistant from two points

Two towns A and B are 6 cm apart on a map. Show the locus of points equidistant from A and B.

Equidistant from two points means the perpendicular bisector of AB.

Open the compasses to more than 3 cm — more than half of AB — and draw arcs above and below the line from A. Keeping the same radius, repeat from B.

Join the two points where the arcs cross. That line is the locus.

Leave the arcs on the diagram: they are what earn the method marks.

Example 2: A region satisfying two conditions

A goat is tethered in a field at point P by a 5 m rope. A fence runs along one edge. Shade the region the goat can reach that is also more than 2 m from the fence.

The first condition gives a circle of radius 5 m about P, drawn to scale with compasses.

The second gives a line parallel to the fence, 2 m away from it on the field side.

Shade the part of the circle lying beyond that parallel line.

Both boundaries must be drawn accurately before any shading, and the shaded region should be labelled so the examiner knows what it represents.

Example 3: Combining a construction with a scale

A map uses a scale of 1 cm to 20 m. A transmitter must be equidistant from two masts that are 140 m apart, and within 100 m of the first mast. Describe how to find the possible positions.

Convert first: 140 m is 7 cm on the map, and 100 m is 5 cm.

Draw the two masts 7 cm apart and construct the perpendicular bisector of the line joining them, which gives the equidistant condition.

Draw a circle of radius 5 cm about the first mast for the distance condition.

The possible positions are the part of the perpendicular bisector lying inside that circle — a segment of the line rather than the whole of it.

Common mistakes and how to avoid them

Rubbing out the construction arcs. They carry the marks. Leave every one.

Using a protractor for a construction. Ruler and compasses only.

Opening the compasses to less than half the segment. The arcs will not meet.

Changing the compass radius partway through a pair of arcs. Keep it fixed.

Shading before drawing all the boundaries. Construct every boundary first, then shade.

Confusing the two "equidistant" loci. Two points give a perpendicular bisector; two lines give an angle bisector.

Forgetting to convert with the scale. Convert into map units before drawing and back into real units afterwards.

Exam technique for "Constructions and Loci"

Read the question to decide which equal-distance condition is being described, then choose the matching construction. That single decision drives the whole answer.

Leave all arcs showing, and use a sharp pencil so lines are thin enough to be accurate.

Draw boundaries for every condition before shading anything, and label the shaded region.

Test one obvious point to confirm which side of a boundary satisfies the condition.

State the scale on a scale drawing and give final answers in real-world units.

Work to about a millimetre and a degree, since these questions carry accuracy marks rather than method marks alone.

Quick revision summary

Every construction is a statement about equal distances, which is why compasses do the work and a protractor is never acceptable. Leave the arcs visible — they carry the marks.

Perpendicular bisector: arcs of equal radius from each end, above and below, joined. It is the locus of points equidistant from two points.

Angle bisector: an arc from the vertex crossing both arms, then equal arcs from those crossings, joined to the vertex. It is the locus of points equidistant from two lines.

Perpendicular from a point to a line: arc from the point crossing the line twice, then bisect that segment.

A 60° angle comes from constructing an equilateral triangle; bisecting gives 30°, and bisecting a right angle gives 45°.

The four basic loci: a circle (fixed distance from a point), a racetrack shape (fixed distance from a line segment), a perpendicular bisector (equidistant from two points), an angle bisector (equidistant from two lines).

For a region, draw every boundary first, decide which side satisfies the condition by testing a point, then shade and label.

On a scale drawing, convert into map units, construct accurately, then convert the measurement back.