A locus is the answer to "where can it be?"
A locus (plural: loci) is the set of all points satisfying a stated condition. A goat tethered to a post by a 5 m rope can stand anywhere within 5 m of the post; the boundary of what it can reach is a circle of radius 5 m, and that circle is the locus of points exactly 5 m from the post.
Every locus question is answered by asking two things: what condition must the point satisfy, and what shape is made by all the points that satisfy it. There are only four standard answers at this level, and every complicated question is a combination of them.
| Condition | Locus | How to construct it |
|---|---|---|
| A fixed distance r from a point A | a circle, centre A, radius r | compasses set to r |
| Equidistant from two points A and B | the perpendicular bisector of AB | equal arcs from A and from B |
| A fixed distance d from a line | two parallel lines, one each side | measure d at two places on each side |
| Equidistant from two intersecting lines | the bisector of the angle between them | arc across both arms, then equal arcs |
Look at d from a line: there are two parallel lines, not one. Drawing only the one above the line is the commonest way to lose a mark in this chapter.
Constructions, and why the arcs must stay
A construction is done with straight edge and compasses only — no protractor and no measuring with a ruler, unless the question says otherwise. The marks are given for the construction arcs, so leaving them on the page is not untidiness, it is the evidence.
These are the constructions the syllabus expects, and each is worth practising until it takes under a minute.
- Perpendicular bisector of AB — arcs of the same radius (more than half of AB) from A and from B, above and below; join the two crossings.
- Bisector of an angle — an arc from the vertex cutting both arms, then equal arcs from those two points; join the vertex to their crossing.
- Perpendicular from a point to a line — an arc from the point cutting the line twice, then bisect the segment between those two cuts.
- Angles of 60° and 90° — 60° from an equilateral triangle construction; 90° from a perpendicular; bisect either to get 30° or 45°.
- Triangles from SSS, SAS or ASA data, and the circumcircle or incircle of a triangle.
Do not erase your working
A perfectly accurate perpendicular bisector drawn with a protractor scores zero in a construction question. The mark is for the pair of arcs, not for the line. Keep them visible, keep the compass radius unchanged when the method requires it, and use a sharp pencil so the crossings are single points.
Regions, and combining two conditions
Questions rarely stop at one condition. "Shade the region that is less than 4 cm from A and nearer to B than to C" needs two loci drawn and then the overlap identified.
The method is mechanical. Draw the boundary for each condition as a line or an arc. Decide which side of each boundary satisfies its condition — a quick test with any convenient point settles it. Then shade only where all the conditions are met.
A and B are 6 cm apart. Shade the region that is within 4 cm of A and nearer to B than to A.
- Draw a circle of radius 4 cm centred on A. The first condition is satisfied inside it."Within" means the interior, and the boundary itself is usually included unless the question says "less than".
- Construct the perpendicular bisector of AB with arcs from A and from B.Nearer to B than to A is decided by the perpendicular bisector — every point on it is equidistant from the two.
- Test which side is nearer to B: pick B itself, which is obviously nearer to B, so the required side is the one containing B.Testing with a convenient point is faster and safer than reasoning about which side is which.
- Shade the part of the circle lying on B's side of the bisector.The answer is the overlap of the two regions, not their union.
The lens-shaped region inside the circle and on B's side of the perpendicular bisector of AB.
Before you leave this chapter
- Fixed distance from a point → circle. Equidistant from two points → perpendicular bisector.
- Fixed distance from a line → two parallel lines, one on each side.
- Equidistant from two lines → the angle bisector.
- Construction arcs are the marks. Never rub them out, and never use a protractor unless invited to.
- For a region, draw each boundary, test which side satisfies the condition, and shade only the overlap.
The contexts these questions actually arrive in
Loci questions are almost never phrased in the language of the chapter. They arrive as a story, and the first task is to translate each sentence into one of the four standard conditions.
| The question says | It means | Draw |
|---|---|---|
| a goat tied by a 5 m rope | at most 5 m from the post | a circle, shade inside |
| equally far from two lighthouses | equidistant from two points | perpendicular bisector |
| at least 20 m from the motorway | a fixed distance from a line | two parallel lines |
| inside the angle between two walls, equally far from each | equidistant from two lines | the angle bisector |
| nearer to A than to B | a region, not a line | the bisector, then shade A's side |
Watch for an obstruction
A goat tied to the corner of a shed does not sweep a full circle: the building blocks part of it, so the region is three quarters of a circle, and if the rope is longer than a wall the animal wraps round the corner with a shorter radius. Reading the question for what blocks the path is worth more than any construction skill.