Loci
Get to grips with loci using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on constructing loci, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. A locus is every point that satisfies a rule, such as a fixed distance from a point.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
Topic overview
A locus is the set of all points satisfying a given condition. The plural is loci, and questions usually ask you to draw the region or path rather than describe it in words.
Four standard loci cover most questions. Points a fixed distance from a point form a circle. Points a fixed distance from a line form a racetrack shape — two parallel lines joined by semicircles at the ends.
Points equidistant from two points form the perpendicular bisector of the line joining them, and points equidistant from two lines form the angle bisector. Constructions and loci therefore go hand in hand.
Revision notes
The four standard loci
Fixed distance from a point: a circle. Fixed distance from a line: parallel lines with semicircular ends.
Equidistant from two points: the perpendicular bisector. Equidistant from two lines: the angle bisector.
Regions rather than lines
If the condition says less than a distance, the locus is a region rather than a line.
Shade the region and use a dashed boundary if the edge is excluded, or a solid one if included — the same convention as with inequalities.
Combining conditions
Questions often give two or more conditions and ask for the region satisfying all of them.
Draw each locus separately, then shade only the overlap. Labelling each construction makes the final region easy to identify.
Key points
- A locus is the set of points meeting a condition.
- Fixed distance from a point gives a circle.
- Fixed distance from a line gives a racetrack shape.
- Equidistant from two points gives the perpendicular bisector.
- Equidistant from two lines gives the angle bisector.
- For several conditions, shade only the overlap.
Worked examples
Example 1
Describe the locus of points \(5\)cm from a fixed point.
Working
Example 2
Describe the locus of points equidistant from two fixed points.
Working
Example 3
Describe the locus of points less than \(3\)cm from a point.
Working
Common mistakes
Drawing a line when a region is required.
Less than or within means a shaded region, not just the boundary.
Forgetting the semicircular ends.
A fixed distance from a line segment includes semicircles at each end.
Erasing the construction arcs.
Loci questions still require the construction evidence.
Shading the wrong overlap.
With several conditions, only the region satisfying all of them counts.
Exam tips
- Draw each locus separately before combining them.
- Use dashed boundaries where the edge is excluded.
- Leave all construction arcs visible.
- Label the final region clearly.
Key terms
- Locus
- The set of all points satisfying a condition.
- Loci
- The plural of locus.
- Equidistant
- The same distance from two things.
- Region
- An area of the diagram satisfying the condition.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.