Sheet № 162 · Foundation + Higher · AQA · Edexcel · OCR
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 exam
§Key definitions
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).
Answer:
Shade the interior of the circle with radius 4 cm centred on T.
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.
§Formulas to memorise
Locus of points a fixed distance from a point = circle with that radius
Locus of points a fixed distance from a line = pair of parallel lines either side, with semicircular ends
Locus of points equidistant from two points = perpendicular bisector of the line segment joining them
Locus of points equidistant from two intersecting lines = angle bisector of the two lines
Read the rule — carefully and identify which standard locus applies.
Construct the locus — using compass and ruler. Leave all construction arcs visible.
If the question asks you to shade a region — , identify which side of the locus satisfies the condition and shade it.
When combining multiple conditions — , construct each locus, then shade only the area where all conditions are satisfied simultaneously.
Worked example
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:
⚠ 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.