Loci in maths

In maths, a locus (plural loci) is the set of all the points that obey a rule, such as "exactly 3 cm from A". Loci maths at GCSE comes down to four standard loci: a circle round a point, a running track shape round a line, the perpendicular bisector of two points and the bisector of an angle. You construct them with a ruler and compasses, then shade the region that fits every rule in the question.

What is a locus in maths?

Think of a locus as every place a point is allowed to be while it keeps to a rule. A goat tied to a post by a 4 m rope can reach every point up to 4 m from the post, so the edge of its grazing is a circle of radius 4 m. The rule decides the shape; your job is to draw it accurately.

AQA's GCSE Maths specification (G2) asks you to use the standard ruler and compass constructions, use them to solve loci problems, and know that the shortest way from a point to a line is the perpendicular. Pearson Edexcel has the same statement in both its Foundation and Higher tier content. These are the words loci questions use:

The 4 loci you need

1. A fixed distance from a point

Set your compasses to the distance, using the scale if there is one, put the point on A and draw a full circle. Points less than 3 cm from A are inside the circle; points more than 3 cm away are outside it.

2. A fixed distance from a line

For a line segment AB, the locus has straight sides parallel to AB, the same distance away, and a semicircle round each end, centred on A and on B. It is often called a running track or stadium shape. If the line goes on forever, such as the edge of a long road, the locus is just the two parallel lines.

Distance to a line is measured at right angles to it, which is why the sides are parallel. Beyond the ends, the nearest part of the segment is the end point itself, so the ends are curved.

3. Equidistant from two points: the perpendicular bisector

1. Open your compasses to more than half the length of AB.
2. Put the point on A and draw arcs above and below the line.
3. Without changing the width, put the point on B and draw arcs that cross the first two, at P and Q.
4. Join P and Q with a ruler. That line is the perpendicular bisector.

It works because P and Q are each one compass width from A and from B, so both are equidistant from A and B, and so is every point on the line through them. Points on A's side of the line are closer to A; points on B's side are closer to B. The same idea is used to find the centre of a circle from two chords.

4. Equidistant from two lines: the angle bisector

1. Put the compass point on the corner O and draw an arc that crosses both arms. Call the crossings X and Y.
2. Put the point on X and draw an arc inside the angle.
3. Keeping the same width, do the same from Y so the two arcs cross.
4. Draw a line from O through the crossing point.

Every point on the bisector is the same perpendicular distance from both arms. Points between the bisector and arm OA are closer to OA; points on the other side are closer to OB.

Perpendiculars and a 60° angle

AQA's specification also lists constructing a perpendicular to a line from or at a given point, and its notes include constructing an angle of 60°.

A perpendicular from a point to a line

1. Put the compass point on the point P and draw an arc that crosses the line twice, at X and Y.
2. From X, then from Y, with the same width, draw arcs that cross on the far side of the line.
3. Join P to that crossing. The new line meets the given line at 90°.

No route from P to the line is shorter than this perpendicular. Any other route is the longest side of a right-angled triangle that has the perpendicular as one of its shorter sides. If P is 4 cm from the line and you go to a point 3 cm along from the foot instead, the distance is √(4² + 3²) = √25 = 5 cm.

A perpendicular at a point on a line

1. With the compass point on P, mark two points X and Y on the line, the same distance either side of P.
2. Open the compasses wider. From X and from Y, draw arcs that cross above P.
3. Join P to the crossing.

An angle of 60°

Draw a line and mark A on it. With the compasses on A, draw a long arc that crosses the line at B. Keep the same width, move the point to B and draw an arc that crosses the first one at C. Join A to C. Angle CAB is 60°, because AB, BC and AC are all one compass width, so triangle ABC is equilateral. Bisect it to get 30°.

Loci and regions

Many questions combine rules and ask you to shade a region. Draw each locus, then decide which side of it you need.

To check you have the right side, pick a test point in your shaded area and ask whether it obeys every rule.

Worked examples

Example 1: a scale drawing

On a map with a scale of 1 cm to 2 km, two villages P and Q are 9 km apart. A new school must be the same distance from P and Q, and no more than 7 km from a station S. How do you find where it can go?

1. Use the scale: 9 km is 9 ÷ 2 = 4.5 cm on the map, and 7 km is 7 ÷ 2 = 3.5 cm.
2. Same distance from P and Q: construct the perpendicular bisector of PQ.
3. No more than 7 km from S: draw a circle of radius 3.5 cm centred on S.
4. The school can go anywhere on the part of the perpendicular bisector inside or on the circle. If the line misses the circle, no place fits both rules.

Example 2: a sprinkler in a garden

A rectangular garden ABCD is 8 m long and 6 m wide. A sprinkler at corner A waters everything within 5 m. Shade the region R that is watered and closer to AB than to AD, using a scale of 1 cm to 1 m.

1. Draw the rectangle 8 cm by 6 cm.
2. Watered means within 5 m of A: set the compasses to 5 cm and draw an arc centred on A. It stays inside the garden because 5 m is less than both 6 m and 8 m.
3. Closer to AB than to AD: bisect the right angle at A. The bisector makes 45° with each side.
4. R is inside the arc and on AB's side of the bisector, so shade that slice.

R is a sector with an angle of 45°, so its area is 45/360 × π × 5² = 9.82 m² to 3 significant figures, about a fifth of the 48 m² garden.

Common mistakes

• Changing the compass width. For a perpendicular bisector, both pairs of arcs must use the same width, or the line won't cut AB in half.
• Arcs too small. Set the compasses to more than half of AB, or the arcs from A and B won't meet.
• Forgetting the round ends. Points a fixed distance from a line segment include a semicircle round each end, not just two parallel lines.
• Mixing up the two bisectors. Equidistant from two points is a perpendicular bisector; equidistant from two lines is an angle bisector.
• Skipping the scale. Convert every real distance to centimetres before you set the compasses.
• Rubbing out your arcs. Construction arcs show your method. Leave them in.
• Shading the wrong side. Test a point in your region against every rule.

In the specification, loci sit alongside bearings and scale drawings, so expect a map or a plan. Try today's GCSE Maths Higher or GCSE Maths Foundation short timed paper. For the angle facts behind the constructions, see perpendicular lines and circle theorems.

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