Loci and Constructions GCSE Maths: Every Compass Technique You Need

Loci and constructions appear in every GCSE Maths exam — and they're one of the topics that separate students who score well from those who drop easy marks. AQA 8300, Edexcel 1MA1, and OCR J560 all test these skills, typically worth 3–5 marks in the Geometry and Measures section of Papers 2 and 3. The method is completely learnable. Follow the steps precisely, leave your arcs visible, and these marks are yours every time.

What You Need to Know First

A locus (plural: loci) is the set of all points that satisfy a given condition — for example, all points 3 cm from a fixed point.
You must use a ruler and compass for all constructions. Freehand curves are not accepted in the exam.
Always leave your construction arcs on the page. Rubbing them out costs you method marks.
The four core constructions are: perpendicular bisector, angle bisector, locus at a fixed distance from a point, and locus at a fixed distance from a line segment.
Most loci questions finish by asking you to shade a region. Read the wording carefully — "within 3 cm", "closer to A than to B", or "between 2 cm and 4 cm" all require different shading.

Perpendicular Bisector Construction

The perpendicular bisector of a line segment crosses it at a right angle exactly at its midpoint. Every point on the perpendicular bisector is equidistant from both ends of the segment — which is why it appears in questions about equidistant positions, boundaries, and planning decisions.

Step-by-step method:

Open your compass to a width greater than half the length of AB.
Place the compass point on A and draw an arc above and an arc below the line.
Without adjusting the compass width, place the point on B and draw two more arcs — they should cross the first pair.
Draw a straight line through the two crossing points. This is the perpendicular bisector.

Worked example 1:

A map shows two towns, A and B, 8 cm apart on the diagram. A new hospital must be built equidistant from both towns. Construct the locus of all possible positions.

Open your compass to approximately 5 cm (more than half of 8 cm).
Draw arcs from A above and below AB.
Keeping the compass the same width, draw arcs from B above and below AB.
The arcs cross at two points, P and Q.
Draw the line through P and Q. Any point on this line is exactly equidistant from A and B.

The hospital can be positioned anywhere along the line PQ.

Worked example 2:

Points A and B are 6 cm apart. Point C lies on the perpendicular bisector of AB, exactly 4 cm above the midpoint M. Find the distance CA.

The midpoint M is 3 cm from A. Point C is 4 cm directly above M.

Using Pythagoras' theorem: CA² = 3² + 4² CA² = 9 + 16 = 25 CA = 5 cm

Angle Bisector Construction

The angle bisector cuts an angle exactly in half. In loci questions, the angle bisector is the set of all points equidistant from two straight lines that meet at a point.

Step-by-step method:

Place the compass point at the vertex (the corner of the angle) and draw an arc that crosses both arms of the angle. Label these crossing points P and Q.
Place the compass point on P and draw an arc inside the angle.
Without changing the compass width, place the point on Q and draw a second arc — it should cross the first arc inside the angle. Label this crossing point R.
Draw a straight line from the vertex through R. This is the angle bisector.

Worked example:

Angle ABC = 72°. A dog is tethered at B and must stay equidistant from the two walls BA and BC. Construct the locus of possible positions.

Place compass at B. Draw an arc crossing BA at P and BC at Q.
Place compass at P. Draw an arc inside the angle.
Same compass width, place at Q. Draw an arc crossing the first arc at R.
Draw the line from B through R.

Every point on line BR is the same perpendicular distance from wall BA and wall BC.

Why does this matter?

Many questions combine the angle bisector with a distance constraint: "within 5 m of B AND closer to BA than BC." Draw the angle bisector first, then draw the circle of radius 5 m, then shade the region that satisfies both conditions.

Locus at a Fixed Distance from a Point or Line

Two patterns come up on every paper.

Pattern 1 — Fixed distance from a single point: the locus is a circle.

A point X moves so it is always 4 cm from a fixed point P. The locus of X is a circle of radius 4 cm centred on P.

In context: a boat must stay at least 3 km from a lighthouse. The boundary of the safe zone is a circle of radius 3 km, and the boat must stay outside it.

Pattern 2 — Fixed distance from a line segment: the locus is a rectangle with semicircular ends.

Worked example:

A security fence must be built exactly 2 cm (on the diagram) from the line segment AB, where AB = 7 cm. Draw the locus of the fence.

Draw a rectangle that is 7 cm long and 2 cm wide on each side of AB — extending 2 cm above AB and 2 cm below AB.
At end A, draw a semicircle of radius 2 cm curving around to the left.
At end B, draw a semicircle of radius 2 cm curving around to the right.

The resulting shape — a rectangle with two semicircular ends — is the complete locus.

A common trap: students draw only the rectangle, forgetting the semicircles at the ends. The semicircles are always required when the question involves a line segment (not an infinite line).

Common Mistakes Students Make

Rubbing out construction arcs. The arcs are the evidence of your method. AQA, Edexcel, and OCR mark schemes all award marks specifically for the arcs — not just the final line. A correct bisector with no arcs may still earn zero method marks.
Changing the compass width mid-construction. On the perpendicular bisector, you must keep identical compass widths for the arcs from A and from B. If you adjust the width between steps, the arcs won't cross at the correct point.
Forgetting semicircles at the ends of line segments. The locus of points at a fixed distance from a line segment always needs semicircular caps. Students who draw just the parallel lines get the rectangle mark but miss the construction mark for the ends.
Shading the wrong region. After drawing your locus boundary, re-read the question. "Within 3 cm of AB" means shade inside the shape. "More than 3 cm from AB" means shade outside. Getting this backwards loses the shading mark.
Using a freehand curve. All arcs must be drawn with a compass. A smoothly drawn freehand arc might look neat, but it won't earn the construction mark.

Exam Tips for Loci and Constructions

Never erase your arcs. This is the single most important rule. The mark scheme for every board explicitly states that arcs must be visible to award the method mark. Erase the arc, lose the mark — regardless of how accurate your final line is.
Use a sharp pencil and check your compass is tight. A loose compass joint can shift the radius mid-arc. Tighten it before the exam or carry a spare.
Label your regions clearly. If asked to shade the required region, shade it lightly and mark it with an R or a tick. Examiners cannot award the mark if they cannot tell which region you intend.
For combined loci, work one condition at a time. Draw condition 1 in full. Then draw condition 2. Then identify the overlap. Students who try to do both at once often miss one constraint entirely.
On Edexcel 1MA1, constructions are checked to ±2 mm. Use the ruler to check your bisector passes through the midpoint, and that your arcs are symmetric about the line.

Practise Now

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