GCSE Maths • Foundation & Higher tier
GCSE Constructions and Loci Revision Guide
Use exact geometric constructions and turn distance conditions into accurate loci.
Foundation
Exam practice
Worked examples
What You'll Learn
Use Construction Equipment
Use a sharp pencil, ruler or straightedge and compasses accurately.
Construct Perpendicular Bisectors
Create a 90° line through the exact midpoint of a segment.
Construct Angle Bisectors
Divide an angle exactly using intersecting compass arcs.
Construct Triangles
Use side lengths and/or angles to locate vertices accurately.
Recognise Standard Loci
Match common distance conditions to circles, parallel lines and bisectors.
Solve Combined Loci Problems
Construct boundaries and select the region satisfying every condition.
Key Rules
Lesson Modules
Bisectors
Construct perpendicular and angle bisectors and use their equidistance properties.
Loci Exam Questions
Translate locus conditions into boundaries, regions and exam-style solutions.
Core Subtopics
Worked Examples
A Region Satisfying Two Locus Conditions
QUESTION
Points A and B are marked on a map. Shade the region that is within 4 cm of A and closer to A than to B.
- Given information: “Within 4 cm” gives a circular region. “Closer to A than B” uses the perpendicular bisector of AB as a boundary.
- Method: Construct both boundaries accurately, identify the valid side or interior for each condition, then take their overlap.
- Set the compass to 4 cm and draw a circle centred at A.
- Construct the perpendicular bisector of AB using equal-radius arcs from A and B.
- The points within 4 cm of A lie inside the circle.
- The points closer to A than B lie on the A side of the perpendicular bisector.
- Shade only the part inside the circle that is also on the A side of the bisector.
- Because “closer” is strict, points on the perpendicular bisector are not part of that condition; follow the diagram convention required by the question.
- Final answer: The required region is the intersection of the circle’s interior and the half-plane on A’s side of the perpendicular bisector.
- Independent answer check: Choose a test point in the shading: its distance from A must be no more than 4 cm and smaller than its distance from B.
- One common mistake: Do not shade the entire circle; the closer-to-A condition removes the part on B’s side.
- Common GCSE Mistakes
- Erasing Construction Arcs
- Changing Compass Width
- Using Too Small a Radius
- Measuring Instead of Constructing
- Shading Between Parallel Loci
- Using Only One Angle Bisector
- Ignoring Boundary Words
- Taking the Union Instead of the Overlap
- Exam Tips
- Use a sharp pencil and keep the compass setting steady.
- Make construction arcs long enough to intersect clearly.
- Measure distance from a line at 90°.
- Label centres, vertices and boundaries so the reasoning is easy to follow.
- Translate each condition separately before combining regions.
- Use a test point to decide which side of a boundary is valid.
- Check whether the object is a point, infinite line, ray or finite segment.
- Do not remove valid construction evidence when finishing the diagram.
Ready to Practice?
Printable Worksheet
Practise bisectors, triangle constructions and standard loci.
Interactive Quiz
Match conditions to circles, parallels, bisectors and regions.
Exam-Style Questions
Construct and shade regions satisfying multiple conditions.
Watch the Constructions and Loci Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Frequently Asked Questions
A locus is the set of all points satisfying a stated condition.
It is the perpendicular bisector of the segment joining the points.
It consists of the two angle bisectors when the full lines are considered; a question’s region may require only one.
It is a circle with that point as centre and the fixed distance as radius.
For an infinite line, it is a pair of parallel lines at that perpendicular distance, one on each side.
Draw both boundaries, identify the valid region for each, and take the overlap when both conditions must be true.
No. Leave them visible unless instructed otherwise, because they demonstrate the construction method.
