EnglandMathsSyllabus dot point

How do you carry out standard ruler-and-compass constructions and use them to find loci and regions?

Standard constructions with ruler and compasses (perpendicular bisector of a line, perpendicular from a point, angle bisector), constructing triangles, and finding loci and regions satisfying given conditions.

A focused answer to the Edexcel GCSE Mathematics geometry content on constructions and loci, covering the standard ruler-and-compass constructions, constructing triangles, and finding loci and regions that satisfy given conditions.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The standard constructions
Constructing triangles
Loci
Combining loci into a region

What this dot point is asking

Edexcel expects you to carry out the standard ruler-and-compass constructions accurately, leaving your construction arcs visible, and to find loci: the sets of points satisfying given distance or equidistance conditions, often combining several to shade a region. These questions reward neat, accurate work and a clear understanding of what each construction produces.

The standard constructions

Each construction uses only a ruler (straight edge) and a pair of compasses, and the arcs must be left on the page as evidence of method.

The perpendicular bisector cuts a line in half at right angles. The angle bisector splits an angle into two equal halves. The perpendicular from a point to a line drops a right-angled line from the point onto the line.

Constructing triangles

A triangle can be constructed accurately from given information. If three sides are given (SSS), draw the base with a ruler, then use compasses set to each of the other two lengths to find the third vertex where the arcs cross. If two sides and the included angle are given (SAS), draw the angle with a protractor and mark the two sides. Accuracy within a small tolerance is expected, and construction arcs should remain visible.

Loci

A locus is the path or region traced by all points obeying a rule.

Find a region from two conditions

A goat is tied in a field at point $G$ with a rope of length $5\,\text{m}$ , and must also stay more than $2\,\text{m}$ from a straight fence $FH$ . Describe and construct the region the goat can reach.

step 1 Handle the rope condition

Within $5\,\text{m}$ of $G$ is the inside of a circle of radius $5\,\text{m}$ centred on $G$ .

step 2 Handle the fence condition

"More than $2\,\text{m}$ from the fence" excludes a $2\,\text{m}$ -wide strip along $FH$ ; the allowed area is beyond a line drawn $2\,\text{m}$ from and parallel to the fence.

step 3 Combine the conditions

The region the goat can reach is inside the circle AND more than $2\,\text{m}$ from the fence, so it is the part of the circle on the far side of the parallel line.

step 4 Shade the overlap

Shade the area that satisfies both conditions at once; this overlap is the required region.

Combining loci into a region

Many exam questions ask for a region satisfying two or three conditions, such as "nearer to $A$ than $B$ " (one side of the perpendicular bisector) and "within $4\,\text{cm}$ of $C$ " (inside a circle). Construct each boundary, decide which side of each satisfies the condition, then shade the area meeting all of them. The construction arcs and boundary lines must be shown, and it helps to lightly test one point to confirm you are shading the correct overlap rather than its opposite.

Exam-style practice questions

Practice questions written in the style of Pearson Edexcel exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Edexcel 20182 marksDescribe the locus of all points that are exactly

3\,\text{cm}

from a fixed point

P

. (Paper 1, non-calculator.)

Show worked answer →

A locus is the set of all points satisfying a condition. Points a fixed distance from a single point form a circle.

The locus is a circle of radius $3\,\text{cm}$ centred on $P$ .

Markers award a mark for "circle" and a mark for the correct radius and centre. Describing it as a single arc, or giving the wrong centre, loses marks. The construction is done with compasses set to $3\,\text{cm}$ .

Edexcel 20213 marksConstruct the perpendicular bisector of a line segment

AB

of length

8\,\text{cm}

, using ruler and compasses only. Describe what the perpendicular bisector represents. (Paper 1, non-calculator.)

Show worked answer →

Open the compasses to more than half of $AB$ . From $A$ , draw arcs above and below the line; from $B$ , draw arcs with the same radius so they cross the first pair. Join the two crossing points with a straight line.

The perpendicular bisector is the locus of points equidistant from $A$ and $B$ .

Markers award marks for visible compass arcs (construction lines must be left in), an accurate perpendicular line, and the correct description. Rubbing out the arcs is penalised because the method must be shown.

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Sources & how we know this

Pearson Edexcel GCSE (9-1) Mathematics (1MA1) specification — Pearson Edexcel (2015)