How do you carry out standard ruler-and-compass constructions and find loci of points satisfying a condition?
Carry out standard constructions (perpendicular bisector, angle bisector, perpendicular from a point) with ruler and compasses, and find loci of points satisfying a given condition, including in combination.
A focused answer to the Eduqas GCSE Mathematics geometry content on constructions and loci, covering the perpendicular bisector, angle bisector and perpendicular from a point, plus the standard loci and combining conditions to find a region.
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What this dot point is asking
The Eduqas geometry content asks you to carry out the standard ruler-and-compass constructions (the perpendicular bisector of a line, the bisector of an angle, and the perpendicular from or to a point) and to find loci: the set of all points satisfying a given condition. Constructions test accurate compass work and the understanding that each construction has a defining geometric property; loci extend this to regions, often combining two or more conditions. Both appear at both tiers, and Eduqas marks the accuracy of the construction (including the visible arcs) as well as the final answer.
The perpendicular bisector
The perpendicular bisector cuts a line segment in half at a right angle.
The defining property, equidistance from the two endpoints, is what makes the perpendicular bisector the locus of points the same distance from two fixed points. Keeping the radius unchanged for both sets of arcs is essential, and the arcs must be left on the diagram as evidence of the method.
The angle bisector
The angle bisector cuts an angle into two equal halves.
So the angle bisector is the locus of points equidistant from two lines that meet at a point. As with the perpendicular bisector, the property (equal distance from both arms) is what links the construction to loci problems.
The standard loci
A locus is the set of all points satisfying a rule, and four standard loci recur.
Recognising which standard locus a condition describes is the key step. "Within cm of a point" is the inside of a circle; "closer to A than to B" is one side of the perpendicular bisector of AB.
Combining conditions to find a region
Many problems combine two or more loci, and the answer is the region satisfying all of them.
Why constructions and loci matter
Constructions and loci connect geometry to real planning problems: positioning a transmitter within range of two towns, finding the safe zone around a hazard, or siting a path equidistant from two boundaries. Eduqas tests careful, accurate compass work and clear reasoning about regions, and because the arcs and lines are part of the evidence, neat construction with the working left visible is what secures the method marks.
Exam-style practice questions
Practice questions written in the style of WJEC Eduqas exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
Eduqas 20183 marksDescribe how to construct the perpendicular bisector of a line segment AB using only a ruler and compasses, and state the key property of every point on it. (Foundation, Component 1, non-calculator.)Show worked answer →
Open the compasses to more than half the length of AB. With the point on A, draw arcs above and below the line. Keeping the same radius, repeat with the point on B so the arcs cross at two points.
Draw a straight line through the two crossing points: this is the perpendicular bisector.
Every point on the perpendicular bisector is equidistant (the same distance) from A and B.
Markers award marks for keeping the radius the same for both arcs, for joining the intersection points, and for stating the equidistant property. Changing the compass radius between the two arcs is the standard error.
Eduqas 20224 marksA goat is tethered by a 5 m rope to a corner of a rectangular barn that it cannot enter. Describe the locus of the region the goat can reach, including what happens at the barn walls. (Higher, Component 2, calculator.)Show worked answer →
Away from the barn, the goat can reach any point within 5 m of the tether point, so the locus is part of a circle of radius 5 m centred on the corner.
The barn blocks part of this, so the reachable region is a three-quarter circle (270 degrees) of radius 5 m outside the barn.
Where the rope wraps round an adjacent corner of the barn, the remaining rope sweeps a smaller arc beyond that corner.
Markers give marks for the three-quarter circle of radius 5 m, for excluding the barn, and for recognising the smaller arc where the rope bends round the next corner. Drawing a full circle, ignoring the barn, is the common error.
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Sources & how we know this
- WJEC Eduqas GCSE (9-1) Mathematics specification (C300) — WJEC Eduqas (2015)