A complete guide to exact ruler-and-compass constructions — perpendicular and angle bisectors, standard angles — and the four core locus rules used to describe paths, boundaries and regions.
Constructions and loci form the practical, drawing-based side of geometry. A construction is an exact drawing made using only a ruler (straight edge) and a pair of compasses — no protractor allowed — which guarantees perfect accuracy rather than an estimate. A locus (plural: loci) is the set of all points that satisfy a given rule, such as "a fixed distance from a point." Together, these skills let you draw and reason about exact positions, boundaries and regions, and they appear constantly in real-world problems such as safe zones, signal ranges and shortest-path questions.
A geometric construction uses only two tools: a straight edge (ruler, for drawing straight lines only — not for measuring) and a pair of compasses (for drawing arcs and circles of a fixed radius). Construction lines and arcs should always be left visible on the final drawing, since they are the proof that the construction was done accurately rather than estimated. A protractor is never used in a formal construction.
The perpendicular bisector of a line segment is a line that crosses it at 90° through its exact midpoint. To construct it: open the compasses to more than half the length of the segment, draw an arc from each end so the arcs cross above and below the line, then join the two crossing points with a straight line.
The perpendicular bisector construction directly proves a key locus rule: every point on it is exactly the same distance from A as it is from B.
An angle bisector is a line that splits an angle exactly in half. To construct it: place the compass point on the vertex and draw an arc crossing both arms of the angle; then, from each of those two crossing points, draw equal-radius arcs that intersect inside the angle; join the vertex to that intersection point.
The two crossing points on the arms are the same distance from the vertex (since one compass setting was used). The final crossing point is equidistant from both of those points (since equal radii were used again). This symmetry is exactly what makes the resulting line bisect the angle.
To construct a perpendicular from a point to a line (dropping a perpendicular), place the compass on the point and draw an arc crossing the line at two places, then construct the perpendicular bisector of the segment between those two crossing points — it will pass through the original point. To construct a perpendicular at a point on a line, draw equal arcs on either side of the point along the line, then bisect the segment between them in the usual way.
Certain angles can be constructed exactly without a protractor. A 60° angle is constructed by drawing an arc from a point on a line, then, using the same radius, drawing a second arc from where the first arc crosses the line — the line joining the vertex to their intersection makes 60° (this is the basis of constructing an equilateral triangle). A 90° angle is a perpendicular, constructed as in Section 4. Bisecting these gives 30° and 45°, and further combinations give 15°, 75°, 105°, and so on.
A locus is the complete set of points that satisfy one specific rule, usually a rule about distance. Instead of finding a single point, a locus problem asks you to describe or draw an entire path, boundary or region. The word comes from Latin for "place"; the plural is loci.
The locus of points a fixed distance from a single point is a circle, with that point as the centre and the fixed distance as the radius.
The locus of points a fixed distance from a straight line segment is a pair of straight lines running parallel to it (one on each side), with semicircular ends that curve around each endpoint of the segment — giving an overall "stadium" or running-track shape.
The locus of points that are the same distance from two fixed points, A and B, is the perpendicular bisector of the line segment AB — exactly the construction from Section 2.
Two mobile phone masts are located at points A and B. To find every location that is exactly the same distance from both masts, construct the perpendicular bisector of AB — every point on that line satisfies the condition.
The locus of points that are the same distance from two straight lines (that meet at an angle) is the angle bisector of the angle between them — exactly the construction from Section 3.
Real problems often combine two or more loci at once, describing a region rather than just a line. Common phrasings include "within 5 km of a point" (the region inside a circle, not just the circle itself), "closer to A than to B" (the region on A's side of the perpendicular bisector of AB), and "within 3 cm of a line" (the inside of the stadium shape). Where two loci are combined, the answer is typically the overlapping region that satisfies both conditions.
A goat is tied to a post at point P with a 4 m rope, inside a garden. The rope also cannot cross a straight wall 2 m from P. Describe the region the goat can graze.
The full reach is a circle of radius 4 m centred on P. However, since the wall is closer than the rope length, the actual grazing region is the part of that circle on the near side of the wall only — the circle is "cut off" by the wall.
Common errors include: using a protractor in a formal construction (only a ruler and compasses are allowed); rubbing out construction arcs, which removes the evidence of an accurate method; confusing "equidistant from two points" (perpendicular bisector) with "equidistant from two lines" (angle bisector); and describing a region as just a boundary line when the question actually asks for the area inside or outside it (e.g. "within 5 km" needs shading, not just a circle).
| Situation | Locus |
|---|---|
| Fixed distance from a point | A circle |
| Fixed distance from a line segment | A "stadium" shape: parallel lines with semicircular ends |
| Equidistant from two points | The perpendicular bisector of the segment joining them |
| Equidistant from two lines | The angle bisector between them |
Every locus problem is really just asking "which of these four situations does this match?" — and every construction problem is built from the same two core techniques: the perpendicular bisector and the angle bisector.
This article is original EDUSAMBAM educational writing. It is designed as a broad constructions and loci resource covering the core ruler-and-compass techniques and locus rules needed for confident, exact geometric drawing. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: practise the perpendicular bisector and angle bisector constructions with real compasses until they are automatic, then work through the four locus rules using scaled diagrams before attempting combined-region problems.
20 questions covering ruler-and-compass constructions, perpendicular and angle bisectors, and the four core locus rules. Answer every question, then submit to see your score instantly.