Loci and Regions –

Loci questions are a staple of GCSE Maths exams across AQA, Edexcel, and OCR. A locus is the set of all points that satisfy a given rule. You need to know four standard loci, be able to construct them using a compass and ruler, and shade the correct region when multiple conditions are combined. This guide covers each type with worked examples and exam-style practice.

What Is a Locus?

A locus (plural: loci) is the path traced by a set of points that obey a particular rule. In GCSE Maths, you are expected to know four standard loci.

Key Formulas

Step-by-Step Method

1. Read the rule carefully and identify which standard locus applies.
2. Construct the locus using compass and ruler. Leave all construction arcs visible.
3. If the question asks you to shade a region, identify which side of the locus satisfies the condition and shade it.
4. When combining multiple conditions, construct each locus, then shade only the area where all conditions are satisfied simultaneously.

Worked Example 1 — Foundation Level

Question: A treasure is buried less than 4 m from a tree T. Show the region where the treasure could be on a scale drawing (1 cm = 1 m).

Working:

The locus of points 4 m from T is a circle of radius 4 cm (at the scale of 1 cm = 1 m). Draw a circle of radius 4 cm centred on T. The treasure is less than 4 m away, so shade the interior of the circle.

Answer: Shade the interior of the circle with radius 4 cm centred on T.

Worked Example 2 — Higher Level

Question: Two points A and B are 6 cm apart. A point P must be closer to A than to B. Construct and shade the region where P can be.

Working:

Construct the perpendicular bisector of AB. All points on the bisector are equidistant from A and B. Points on the A-side of the bisector are closer to A.

Shade the region on the same side as A.

Answer: Shade the region on the A-side of the perpendicular bisector of AB.

Worked Example 3 — Exam Style

Question: A dog is attached to a post P by a lead of length 5 m. A straight wall runs 3 m from P. Using a scale of 1 cm = 1 m, shade the region the dog can reach.

Working:

Step 1 — Draw the post P and the wall as a straight line 3 cm away.

Step 2 — The dog can reach all points within 5 m (5 cm on the scale) of P, so draw a circle of radius 5 cm centred on P.

Step 3 — The wall blocks the dog's path. On the wall side, the dog can only go as far as the wall, then move along it until the lead runs out. The accessible region is the part of the circle on the near side of the wall, plus curved portions where the lead wraps around the ends of the wall.

Answer: Shade the region that is within 5 cm of P but not beyond the wall, with modified arcs at the wall ends.

Common Mistakes

• Not using a compass for circles. Freehand circles are not acceptable in construction questions. Always use a compass.
• Forgetting semicircular ends. The locus of points a fixed distance from a line segment includes semicircles at each end of the segment, not just two parallel lines.
• Shading the wrong region. Read the inequality carefully — "less than" means shade inside the locus; "more than" means shade outside. "Closer to A than B" means shade the A-side of the perpendicular bisector.
• Drawing straight lines instead of curves for the locus from a line segment. The ends of the locus from a line segment are semicircles, not straight corners.

Exam Tips

• Highlight the key words in the question: "less than", "more than", "equidistant", "closer to".
• When combining loci, draw each one in a different style (solid, dashed) to keep them distinct, then shade only the overlap.
• Construction arcs must be visible — do not erase them.
• Loci questions often combine distance from a point (circle), distance from a line (parallel lines), and equidistant conditions (bisectors). Practise combining all three.
• Use a sharp pencil and keep your compass tight so it does not slip mid-arc.
• If the question says "nearer to A than B", construct the perpendicular bisector of AB and shade the A-side.
• Check your shaded region satisfies every condition stated in the question before moving on.

Practice Questions

Q1 (Foundation): Draw the locus of points that are exactly 3 cm from a fixed point A.

Q2 (Foundation): Two walls meet at a corner. Shade the locus of points equidistant from both walls.

Q3 (Higher): Points A and B are 8 cm apart. Shade the region of points that are closer to A than to B and within 5 cm of A.

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Related Topics

• Perpendicular Bisector and Angle Bisector
• Constructions and Loci
• Bearings
• Scale Drawings and Maps
• Circle Theorems

Summary

• A locus is the set of all points satisfying a given rule. The four standard loci are: a circle (fixed distance from a point), parallel lines with semicircular ends (fixed distance from a line segment), perpendicular bisector (equidistant from two points), and angle bisector (equidistant from two lines). Shade the correct region by reading the inequality carefully. Always construct with compass and ruler and leave arcs visible. When multiple conditions apply, the answer region is the overlap of all individual loci.