How do you carry out ruler-and-compass constructions and find loci that satisfy a condition?
Construct perpendicular bisectors, perpendiculars from a point and angle bisectors with ruler and compasses, and find loci and regions defined by distance conditions, including bearings.
A CCEA GCSE Mathematics answer on constructions and loci, covering ruler-and-compass construction of perpendicular bisectors, perpendiculars and angle bisectors, the standard loci, regions defined by distance conditions, and the link to bearings.
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What this dot point is asking
Constructions and loci use ruler and compasses to draw exact figures and to find the set of points satisfying a condition. In the CCEA Geometry and Measures strand you must construct perpendicular bisectors, perpendiculars from or to a line, and angle bisectors, and find loci and regions defined by distances, including problems set with bearings. Accuracy and leaving the construction arcs visible are part of the marks, because they show the correct method.
What a construction is
A construction is an exact drawing made with a straight edge and a pair of compasses, without measuring angles by protractor. The compass arcs that produce the construction must be left on the diagram, because they are the evidence of the correct method and CCEA credits them. Keep the compass setting fixed where the method requires it.
The standard constructions
Three constructions are examined, each built from intersecting arcs.
The perpendicular bisector passes through the midpoint at a right angle; the angle bisector splits the angle into two equal halves.
Loci
A locus is the set of all points that satisfy a given condition. The standard loci come straight from the constructions.
Regions and combined conditions
Many questions ask for a region satisfying more than one condition, such as "within 4 cm of and closer to line than ".
Bearings
A bearing is a direction measured clockwise from north, always written with three figures, so a direction of is written . Bearings combine with the angle rules and with trigonometry to solve navigation problems, and back bearings differ by . Drawing the north line at each point is the reliable way to read a bearing correctly.
Why this matters
Constructions and loci develop precise drawing and spatial reasoning, and they appear in practical contexts such as planning, mapping and design. They link to the angle rules, to circles, and to bearings with trigonometry, and CCEA rewards accurate work with visible construction arcs. Treating the arcs as part of the answer, not rough work, is what secures the marks.
Exam-style practice questions
Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
CCEA 20193 marksDescribe the locus of points that are exactly cm from a fixed point , and the locus of points equidistant from two fixed points and . (Non-calculator.)Show worked answer →
A fixed distance from a single point is a circle.
The locus of points cm from is a circle of radius cm centred on .
Points equidistant from two points lie on the perpendicular bisector of the line joining them.
So the second locus is the perpendicular bisector of . Marks are for the circle (with radius and centre), and for naming the perpendicular bisector. Describing a region instead of a line, or omitting the radius, lose marks.
CCEA 20212 marksA ship sails on a bearing of from port. State what a bearing measures and from where. (Non-calculator.)Show worked answer →
A bearing is an angle measured clockwise from north, written as three figures.
So means measured clockwise from the north line at the starting point.
One mark is for "clockwise from north" and one for the three-figure convention. Measuring anticlockwise, or from east, is the usual misunderstanding.
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Sources & how we know this
- CCEA GCSE Mathematics specification (2210) — CCEA (2017)