Constructions and Loci –

Constructions and loci is a practical geometry topic tested at both Foundation and Higher tiers on AQA, Edexcel, and OCR GCSE Maths papers. You need to know how to use a pair of compasses and a straight edge (ruler) to construct perpendicular bisectors, angle bisectors, and other standard constructions accurately. You also need to understand loci — the set of points that satisfy a given rule — and shade or draw regions based on multiple conditions. This guide covers every construction and locus type you may be asked about, with clear step-by-step instructions, worked examples, and practice questions. For related geometry, see our angles and parallel lines guide.

What Are Constructions and Loci?

Constructions

A construction is an accurate geometric drawing made using only a pair of compasses and a straight edge. You must leave your construction arcs visible — rubbing them out loses marks because the examiner needs to see that you used the correct method.

Loci

A locus (plural: loci) is the set of all points that satisfy a particular condition. For example, the locus of points that are exactly 3 cm from a fixed point is a circle of radius 3 cm centred on that point.

The Four Standard Loci

1. Fixed distance from a point — a circle.
2. Fixed distance from a line — a "racetrack" shape (two parallel lines with semicircular ends).
3. Equidistant from two points — the perpendicular bisector of the line segment joining the two points.
4. Equidistant from two lines — the angle bisector of the angle formed by the two lines.

Step-by-Step Method

Constructing a Perpendicular Bisector

1. Set your compasses to more than half the length of the line segment.
2. Place the compass point on one end of the line and draw arcs above and below.
3. Without changing the compass width, place the compass point on the other end and draw arcs above and below.
4. The arcs intersect at two points. Draw a straight line through both intersection points.
5. This line is the perpendicular bisector — it crosses the original line at 90° and cuts it exactly in half.

Constructing an Angle Bisector

1. Place the compass point on the vertex of the angle and draw an arc that crosses both arms of the angle.
2. Place the compass point on one intersection and draw an arc between the two arms.
3. Without changing the compass width, place the compass point on the other intersection and draw another arc.
4. Draw a straight line from the vertex through the point where the two arcs cross.
5. This line bisects the angle — it divides it into two equal parts.

Constructing a Perpendicular from a Point to a Line

1. Place the compass point on the given point and draw an arc that crosses the line in two places.
2. From each crossing point, draw arcs below the line (using the same radius) so they intersect.
3. Draw a straight line from the given point through the intersection of the arcs. This line meets the original line at 90°.

Constructing a 60° Angle

1. Draw a straight line and mark a point on it.
2. Place the compass on that point and draw an arc that crosses the line.
3. Without changing the width, place the compass where the arc crosses the line and draw another arc crossing the first.
4. Draw a line from the original point through the intersection. The angle between the two lines is 60°.

Worked Example 1 — Foundation Level

Question: Two mobile phone masts are at points A and B, which are 8 cm apart on a map. The signal from mast A reaches up to 5 cm and the signal from mast B reaches up to 4 cm. Shade the region that receives signal from both masts.

Working:

Step 1 — Draw points A and B, 8 cm apart.

Step 2 — Draw a circle of radius 5 cm centred on A (the locus of points within 5 cm of A).

Step 3 — Draw a circle of radius 4 cm centred on B (the locus of points within 4 cm of B).

Step 4 — The region receiving both signals is where the two circles overlap. Shade this intersection.

Answer: The shaded region is the intersection of the two circles.

Worked Example 2 — Higher Level

Question: A rectangular garden ABCD has AB = 10 m and BC = 6 m. A tree must be planted so that it is closer to AB than to CD, more than 3 m from corner A, and less than 7 m from corner B. Show the region where the tree can be planted.

Working:

Step 1 — Closer to AB than to CD: the locus of points equidistant from AB and CD is the perpendicular bisector of the distance between them — a horizontal line 3 m from each. The tree must be below (nearer AB) this line.

Step 2 — More than 3 m from A: draw a circle of radius 3 m centred on A. The tree must be outside this circle.

Step 3 — Less than 7 m from B: draw a circle of radius 7 m centred on B. The tree must be inside this circle.

Step 4 — The valid region is the intersection of all three conditions. Shade it.

Answer: The region satisfying all three conditions is shaded within the garden boundary.

Common Mistakes

• Rubbing out construction arcs. This is the most common error. Leave all arcs visible — they prove you used the correct construction method.
• Setting the compass width too small. For a perpendicular bisector, the compass must be set to more than half the line length, otherwise the arcs will not intersect.
• Inaccurate compass work. Make sure the compass point does not slip. Hold it firmly at the centre and rotate smoothly.
• Confusing "closer to" with "equidistant from". "Equidistant" means on the bisector line. "Closer to A than B" means on A's side of the perpendicular bisector.
• Not reading all conditions. Loci questions often have two or three constraints. Shade only the region satisfying all of them simultaneously.

Exam Tips

• Bring a sharp pencil and a reliable pair of compasses. Accuracy matters — marks can be lost if constructions are more than 2 mm out.
• Label your constructions. If the question asks for a perpendicular bisector, write "perpendicular bisector" next to it.
• For combined loci questions, draw each locus separately first (in pencil), then identify and shade the region satisfying all conditions.
• Use a ruler to check distances after drawing, to verify your construction is accurate.
• Practise with real equipment. This is one of the few topics where physical practice (as opposed to just doing calculations) makes a significant difference.

Practice Questions

Question 1: Construct the perpendicular bisector of a line segment 7 cm long.

Question 2: Construct the bisector of a 70° angle.

Question 3: Draw the locus of points exactly 4 cm from point P.

Question 4: Two points X and Y are 6 cm apart. Shade the locus of points that are closer to X than to Y.

Question 5: A dog is tied to a stake by a 5 m rope. On one side there is a wall 3 m from the stake, parallel to the stake. Describe the shape of the region the dog can reach.

Build your construction and loci skills at GCSEMathsAI — our AI tutor can guide you through each step and mark your working.

Related Topics

• Bearings — accurate drawing with angles measured from north.
• Angles: Basic Rules and Parallel Lines — angle measurement fundamentals.
• Circle Theorems — properties of circles relevant to loci.
• Plans and Elevations — another topic requiring accurate drawing.

Summary

Constructions and loci combine practical drawing skills with geometric reasoning. You must be able to construct perpendicular bisectors, angle bisectors, and perpendiculars using only compasses and a straight edge — and always leave your arcs visible. Loci questions ask you to draw or shade the set of points satisfying given conditions, often combining circles (fixed distance from a point), perpendicular bisectors (equidistant from two points), and angle bisectors (equidistant from two lines). Accuracy, clear labelling, and working with real equipment are the keys to full marks on this topic.