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 OCR GCSE Mathematics geometry content on constructions and loci, covering the perpendicular bisector, angle bisector and perpendicular from a point, and finding loci of points satisfying a condition.
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What this dot point is asking
OCR reference G2 covers standard ruler-and-compass constructions and loci. A construction builds an exact figure using only a straight edge and compasses, and a locus is the set of all points satisfying a given condition. These are practical, accuracy-rewarding topics where OCR awards marks for the visible construction arcs, not just the final line. They appear on every tier and link to bearings, scale drawing and the circle.
Standard constructions
Three constructions recur in the exam.
The golden rule is to keep the compass radius fixed within each step and to leave every arc on the page. OCR's mark scheme credits a correct method even if the final line is slightly off, so erasing the arcs throws away marks. Accuracy to within about mm and is expected.
What a locus is
A locus generalises "the set of all points where ...".
So "all points exactly cm from a post" is a circle of radius cm, and "all points equidistant from two fences meeting at a corner" is the angle bisector of that corner. Recognising which standard locus a worded condition describes is the key step, after which the construction follows.
Loci can also describe inequalities rather than exact distances. "Within cm of a post" is the region inside the circle, and "nearer to wall than wall " is the region on one side of the angle bisector. Exam questions usually ask you to shade such a region rather than just draw a line, so read whether the condition is an exact distance (a boundary) or a "less than" or "more than" (a region). The accurate construction of the boundary always comes first, then the correct side or interior is shaded.
Combining loci
Many questions intersect two conditions to find a region.
Why constructions and loci matter
Constructions train precise geometric technique, and loci model real "where can it reach" or "which area is safe" problems, such as a tethered animal, a signal range, or a path equidistant from two roads. OCR rewards the accurate, arc-showing method and the correct identification of each standard locus. Combining loci to shade a region is the most common exam form and tests the AO3 ability to translate a worded condition into geometry.
Exam-style practice questions
Practice questions written in the style of OCR exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
OCR 20183 marksUsing ruler and compasses only, construct the perpendicular bisector of a line segment of length cm. Show all your construction arcs. (Foundation, Paper 1, calculator.)Show worked answer →
Open the compasses to more than half the length of (more than cm).
With the point on , draw arcs above and below the line; without changing the radius, repeat with the point on . The arcs cross at two points.
Draw a straight line through the two crossing points: this is the perpendicular bisector.
Markers award a mark for a consistent radius greater than half , a mark for all four arcs visible, and a mark for the bisector line. Erasing the construction arcs loses a mark, because OCR awards marks for the method shown.
OCR 20214 marksA goat is tethered by a m rope to a corner of a rectangular barn. Describe and sketch the locus of points the goat can reach in the open field outside the barn. (Higher, Paper 4, calculator.)Show worked answer →
The locus of points a fixed distance from a point is a circle (or arc) of that radius.
Near the tethering corner, the goat sweeps a three-quarter circle of radius m in the open field (the barn blocks one quarter).
As the rope wraps around each adjacent corner of the barn, the goat can sweep smaller quarter-circle arcs, with radius reduced by the length of wall already used.
Markers give marks for the main arc of radius m, for recognising the barn blocks part of it, for the wrap-around arcs, and for a clear sketch. Drawing a full circle, ignoring the barn, is the standard error.
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Sources & how we know this
- OCR GCSE (9-1) Mathematics (J560) specification — OCR (2015)