Use compasses and a ruler to construct bisectors, then combine circles, lines and bisectors to shade regions defined by distance rules. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A locus is the path a point follows under a rule. Fixed distance from a point: a circle. Equidistant from two points: the perpendicular bisector. Equidistant from two lines: the angle bisector. A region 'closer to A than B' is the side of the perpendicular bisector that contains A.
Constructions with compasses: perpendicular bisector, angle bisector, a perpendicular from a point to a line, an equilateral triangle. Leave the arcs — they are the method marks. A shading question is two loci and the overlap.
Locus of a distance from a point is a circle; from a line is two parallels (plus end-caps if the line is a segment). Equidistant from two points is the perpendicular bisector; from two lines is the angle bisector. Constructions: compass width locked, two arcs, join.
Objectives
Points a fixed distance from one point form a circle.
The locus is the perpendicular bisector of the segment joining them.
The locus is the angle bisector.
Key points
Construct the perpendicular bisector of a line segment with compasses only.
Construct the bisector of an angle.
Know the four basic loci: fixed distance from a point, from a line, equidistant from two points, equidistant from two lines.
Use a scale to turn real distances into drawing distances.
Worked examples
Example 1
A point P is always 2 cm from a fixed point A. P moves to make a complete circle. What is the diameter of this circle in centimetres?
1 marks
P is always 2 cm from A, so the radius is 2 cm.
Diameter = 2 × radius = 2 × 2.
The diameter is 4 cm.
Answer: 4
Common slips
Slip: P is always 4 cm from A, so the radius is 2 cm. That changes a number from the question, so it answers a different question.
Slip: Diameter = 5 × radius = 2 × 2. That changes a number in the working, so the step no longer follows from the one before.
Slip: The diameter is 8 cm. That does not match the working above it.
Example 2
A and B are 12 cm apart. The perpendicular bisector of AB crosses AB at M. How far is M from A?
1 marks
M is the midpoint of AB, and AB is 12 cm long.
12 ÷ 2 = 6.
M is 6 cm from A.
Answer: 6
Common slips
Slip: M is the midpoint of AB, and AB is 14 cm long. That changes a number from the question, so it answers a different question.
Slip: 12 × 2 = 24. That uses multiplication, but this step needs division.
Slip: M is 10 cm from A. That does not match the working above it.
Example 3
A and B are 6 cm apart. P is on the perpendicular bisector of AB and is 4 cm from the midpoint M of AB. How far is P from A?
3 marks
M is the midpoint, so AM = 6 ÷ 2 = 3 cm.
PA² = 3² + 4² = 9 + 16 = 25.
PA = √25 = 5 cm.
Answer: 5
Common slips
Slip: M is the midpoint, so AM = 8 ÷ 2 = 3 cm. That changes a number from the question, so it answers a different question.
Slip: PA² = 6² + 4² = 9 + 16 = 25. That changes a number in the working, so the step no longer follows from the one before.
Slip: PA = √29 = 5 cm. That does not match the working above it.
Questions and answers
GCSE Maths loci and constructions questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A and B are 10 cm apart. P is on the perpendicular bisector of AB and is 12 cm from the midpoint M of AB. How far is P from A?
Show answer
Answer: 13
M is the midpoint, so AM = 10 ÷ 2 = 5 cm.
PA² = 5² + 12² = 25 + 144 = 169.
M1 — M is the midpoint, so AM = 10 ÷ 2 = 5 cm.
A1 — Answer 13
Question 2 Easy2 marks
Angle ABC is 70°. The angle bisector of ABC is drawn. What is the size of each of the two equal angles it makes?
Show answer
Answer: 35
Angle ABC = 70° is split into two equal angles.
70° ÷ 2 = 35°.
M1 — Angle ABC = 70° is split into two equal angles.
A1 — Answer 35
Question 3 Medium2 marks
The point P is the same distance from A and from B. P is 5 cm from A. How far is P from B?
Show answer
Answer: 5
P is the same distance from A and B, and it is 5 cm from A.
Every point on the perpendicular bisector of AB is equally far from A and B.
M1 — P is the same distance from A and B, and it is 5 cm from A.
A1 — Answer 5
Question 4 Medium2 marks
A and B are 12 cm apart. P is on the perpendicular bisector of AB and is 8 cm from the midpoint M of AB. How far is P from A?
Show answer
Answer: 10
M is the midpoint, so AM = 12 ÷ 2 = 6 cm.
PA² = 6² + 8² = 36 + 64 = 100.
M1 — M is the midpoint, so AM = 12 ÷ 2 = 6 cm.
A1 — Answer 10
Question 5 Hard2 marks
A and B are 16 cm apart. The perpendicular bisector of AB crosses AB at M. How far is M from A?
Show answer
Answer: 8
M is the midpoint of AB, and AB is 16 cm long.
16 ÷ 2 = 8.
M1 — M is the midpoint of AB, and AB is 16 cm long.
A1 — Answer 8
Question 6 Hard2 marks
The point P is the same distance from A and from B. P is 12 cm from A. How far is P from B?
Show answer
Answer: 12
P is the same distance from A and B, and it is 12 cm from A.
Every point on the perpendicular bisector of AB is equally far from A and B.
M1 — P is the same distance from A and B, and it is 12 cm from A.
A1 — Answer 12
Question 7 Exam-style3 marks
A and B are 8 cm apart. P is on the perpendicular bisector of AB and is 3 cm from the midpoint M of AB. How far is P from A?
Show answer
Answer: 5
M is the midpoint, so AM = 8 ÷ 2 = 4 cm.
PA² = 4² + 3² = 16 + 9 = 25.
PA = √25 = 5 cm.
M1 — M is the midpoint, so AM = 8 ÷ 2 = 4 cm.
M1 — PA² = 4² + 3² = 16 + 9 = 25.
A1 — Answer 5
Common mistakes
Most lost marks on this topic are one slip.
Drawing one point instead of the whole circle.
The locus is every point at that distance.
Using the angle bisector when the condition is about two points.
Two points need the perpendicular bisector.
Rubbing out the construction arcs.
The arcs show how the line was found.
Spot the slip
The set of points 4 cm from A is
a circle with centre A and radius 4 cm
a straight line 4 cm long
the point 4 cm above A
Show the slip
a circle with centre A and radius 4 cm Every point on that circle is 4 cm from A.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Use a sharp pencil and press lightly; accuracy is usually judged within 2 mm and 2 degrees.
Write the scaled distance on your diagram (e.g. '5 m = 5 cm') so the examiner can follow it.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.