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GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 3–6 · Geometry

Constructions and Loci

With only a ruler and a pair of compasses you can bisect a line, bisect an angle and drop a perpendicular. Loci questions then ask where something can be, such as a goat on a rope or a phone signal. Examiners need to see your construction arcs, so never rub them out.

1The golden rules

  • Use a sharp pencil and keep the compass width fixed for each pair of arcs.
  • Leave all construction arcs visible. Without them you get no marks, even if the line is in the right place.
  • Draw arcs long enough to cross each other clearly.

2Perpendicular bisector of a line segment

  1. Open the compasses to more than half the length of AB.
  2. With the point on A, draw arcs above and below the line.
  3. Keep the same width. With the point on B, draw arcs that cross the first two.
  4. Join the two crossing points with a straight line.
AB
The perpendicular bisector cuts AB in half at 90°
Key ideaEvery point on the perpendicular bisector of AB is the same distance from A and B.

3Bisecting an angle

  1. With the compass point on the vertex O, draw an arc crossing both arms of the angle.
  2. From each crossing point, draw an arc (same width) so the two arcs cross inside the angle.
  3. Draw a line from O through the crossing point.
O
The bisector splits the angle into two equal halves
Key ideaEvery point on the angle bisector is the same distance from both arms.

4The perpendicular from a point to a line

  1. With the compass point on P, draw an arc that cuts the line in two places.
  2. Construct the perpendicular bisector of those two points.
  3. It passes through P and meets the line at 90°.
P
Exam tipThe shortest distance from a point to a line is always the perpendicular distance.

5The four standard loci

A locus (plural loci) is the set of all points that follow a rule.

P
Fixed distance from a point: a circle
Fixed distance from a line segment: two parallel lines joined by semicircles
AB
Equidistant from two points: the perpendicular bisector
Equidistant from two lines: the angle bisector

6Regions and scale

Many questions combine two rules and ask you to shade the region where both are true. If there is a scale, convert real distances first.

Worked example
A field ABCD is 10 m by 6 m. A goat is tied to corner A with a 4 m rope. The farmer only wants it to graze where it is nearer to AB than to AD. Show the region and find its area.
  1. Within 4 m of A: a quarter circle of radius 4 m centred at A.
  2. Nearer AB than AD: below the bisector of angle DAB (the 45° line).
  3. The region is a 45° sector of radius 4 m: area = 45360 × π × 4² = 2π.
≈ 6.28 m²
ABDC
Shaded: within 4 m of A and nearer to AB than to AD
Common mistakes
  • Rubbing out construction arcs. Leave them all visible.
  • Changing the compass width between the two arcs of a pair.
  • Drawing the locus round a line segment as a rectangle. The ends should be semicircles.
  • Forgetting to convert real distances using the scale before drawing.
  • Shading the wrong side of a boundary. Re-read whether the question says nearer, further, within or at least.

★Key Vocabulary Recap

Locus
The set of points that satisfy a rule.
Perpendicular bisector
The line cutting a segment in half at right angles.
Angle bisector
The line cutting an angle into two equal angles.
Equidistant
The same distance from two things.
Region
An area made of all points satisfying one or more conditions.
Specification reference (Pearson Edexcel GCSE (9–1) Mathematics 1MA1: G2, G15)
  • G2: use the standard ruler and compass constructions (perpendicular bisector of a line segment, perpendicular to a line from or at a given point, angle bisector).
  • G2: know that the perpendicular distance from a point to a line is the shortest distance; use these to construct given figures and solve loci problems.
  • G15: interpret scale drawings and maps.
Next stepReady to practise? Try the 12 exam-style questions, then check your working against the worked solutions.
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Practice questions

GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 3–6 · Geometry

Constructions and Loci — Practice Questions

12 exam-style questions (27 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Foundation · Grades 3–4
  1. (1 mark)
    Grade 3
    Describe the locus of all points that are 3 cm from a fixed point P.
  2. (1 mark)
    Grade 3
    Describe the locus of all points that are the same distance from point A as from point B.
  3. (2 marks)
    Grade 4
    AB is a line 8 cm long. Describe how to construct the perpendicular bisector of AB using ruler and compasses.
  4. (2 marks)
    Grade 4
    An angle measures 70°. Describe how to bisect it using ruler and compasses, and state the size of each new angle.
Foundation and Higher · Grades 4–5
  1. (3 marks)
    Grade 5
    A straight fence is 6 m long. A security light detects anything within 2 m of the fence. Describe the shape of the region it detects and find its perimeter. Give your answer to 1 decimal place.
  2. (3 marks)
    Grade 5
    A goat is tied to the corner of a large rectangular field with a 6 m rope. Find the area of grass it can reach. Give your answer to 3 significant figures.
  3. (3 marks)
    Grade 5
    A goat is tied to the middle of a long straight wall with a 5 m rope. Find the area it can reach, to 3 significant figures.
  4. (2 marks)
    Grade 5
    A map has scale 1 cm : 2 km. A phone mast has a range of 6 km. Describe what you would draw on the map to show the area it covers.
Higher · Grade 6
  1. (2 marks)
    Grade 6
    Describe how to construct the shortest path from a point P to a straight line, using ruler and compasses only.
  2. (2 marks)
    Grade 6
    AB = 6 cm and M is the midpoint of AB. How many points are both equidistant from A and B and exactly 2 cm from M? Explain.
  3. (3 marks)
    Grade 6
    A field ABCD is a 10 m by 6 m rectangle. A goat tied at A on a 4 m rope must stay nearer to side AB than to side AD. Find the area of the region it can graze, to 3 significant figures.
  4. (3 marks)
    Grade 6
    Explain how to construct an angle of 30° using only a ruler and compasses.
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
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Worked solutions

Show worked solutions
GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 3–6 · Geometry

Constructions and Loci — Worked Solutions

Full worked solutions to all 12 questions (27 marks), set out the way an Edexcel examiner wants to see them.

Foundation · Grades 3–4
  1. (1 mark)
    Grade 3
    Describe the locus of all points that are 3 cm from a fixed point P.
    Solution
    1. Every point 3 cm from P lies on a circle.
    A circle, centre P, radius 3 cm
  2. (1 mark)
    Grade 3
    Describe the locus of all points that are the same distance from point A as from point B.
    Solution
    1. Equidistant from two points → the perpendicular bisector.
    The perpendicular bisector of AB
  3. (2 marks)
    Grade 4
    AB is a line 8 cm long. Describe how to construct the perpendicular bisector of AB using ruler and compasses.
    Solution
    1. Set compasses to more than 4 cm. With the point on A, draw arcs above and below AB.
    2. Same width, point on B: draw arcs crossing the first two.
    3. Join the two crossing points. Leave the arcs showing.
    AB
    Arcs from A and B, then join the crossing points
  4. (2 marks)
    Grade 4
    An angle measures 70°. Describe how to bisect it using ruler and compasses, and state the size of each new angle.
    Solution
    1. Arc from the vertex crossing both arms.
    2. Arcs of equal width from the two crossing points, meeting inside the angle.
    3. Join the vertex to the meeting point. Each angle = 70 ÷ 2 = 35°.
    Each angle is 35°
Foundation and Higher · Grades 4–5
  1. (3 marks)
    Grade 5
    A straight fence is 6 m long. A security light detects anything within 2 m of the fence. Describe the shape of the region it detects and find its perimeter. Give your answer to 1 decimal place.
    Solution
    1. Region: two straight sides 6 m long, parallel to the fence, joined by two semicircles of radius 2 m.
    2. Perimeter = 2 × 6 + (two semicircles = one full circle) 2π × 2 = 12 + 4π.
    3. = 24.566…
    24.6 m
  2. (3 marks)
    Grade 5
    A goat is tied to the corner of a large rectangular field with a 6 m rope. Find the area of grass it can reach. Give your answer to 3 significant figures.
    Solution
    1. At a corner the goat can reach a quarter circle of radius 6 m.
    2. Area = 14 × π × 6² = 9π = 28.27…
    28.3 m²
  3. (3 marks)
    Grade 5
    A goat is tied to the middle of a long straight wall with a 5 m rope. Find the area it can reach, to 3 significant figures.
    Solution
    1. The wall blocks one side, so the region is a semicircle of radius 5 m.
    2. Area = ½ × π × 5² = 12.5π = 39.26…
    39.3 m²
  4. (2 marks)
    Grade 5
    A map has scale 1 cm : 2 km. A phone mast has a range of 6 km. Describe what you would draw on the map to show the area it covers.
    Solution
    1. 6 km ÷ 2 = 3 cm on the map.
    2. Draw a circle of radius 3 cm centred on the mast and shade inside it.
    Circle of radius 3 cm
Higher · Grade 6
  1. (2 marks)
    Grade 6
    Describe how to construct the shortest path from a point P to a straight line, using ruler and compasses only.
    Solution
    1. Arc centred P cutting the line twice.
    2. Construct the perpendicular bisector of those two points; it passes through P.
    3. The path is along this perpendicular.
    P
    Construct the perpendicular from P to the line
  2. (2 marks)
    Grade 6
    AB = 6 cm and M is the midpoint of AB. How many points are both equidistant from A and B and exactly 2 cm from M? Explain.
    Solution
    1. Equidistant from A and B: the perpendicular bisector of AB, which passes through M.
    2. Points 2 cm from M: a circle radius 2 cm about M.
    3. The line crosses the circle twice.
    2 points
  3. (3 marks)
    Grade 6
    A field ABCD is a 10 m by 6 m rectangle. A goat tied at A on a 4 m rope must stay nearer to side AB than to side AD. Find the area of the region it can graze, to 3 significant figures.
    Solution
    1. Within 4 m of A: a quarter circle (90°).
    2. Nearer AB than AD: on the AB side of the angle bisector, so half of the quarter circle → a 45° sector.
    3. Area = 45360 × π × 16 = 2π = 6.283…
    ABDC
    6.28 m²
  4. (3 marks)
    Grade 6
    Explain how to construct an angle of 30° using only a ruler and compasses.
    Solution
    1. Draw a line and mark a point O. Draw an arc from O cutting the line at P.
    2. With the same width, draw an arc from P cutting the first arc at Q. Triangle OPQ is equilateral, so angle POQ = 60°.
    3. Bisect angle POQ to get 30°.
    Construct 60° (equilateral triangle), then bisect it
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