A goat tied to a post can only graze so far — the edge of where it can reach traces a perfect circle. That grazing boundary is a locus: the set of all points that follow a rule, the answer to "where could it be?". This chapter pairs loci with constructions — accurate drawings made with only a ruler and a pair of compasses, no protractor allowed.
Key constructions
Each of these is built from arcs swept with compasses. The magic word to remember is equidistant — meaning "equal distance from".
- Perpendicular bisector of a line AB — the points equidistant from A and B. Drawn with two equal arcs swung from A and from B; join where they cross.
- Angle bisector — splits an angle into two equal halves; it is the set of points equidistant from the two lines that form the angle.
- Perpendicular from a point to a line — the shortest, right-angled distance from the point to the line.
Common loci
Most locus questions are built from just three basic pictures. Learn the shape each rule produces:
- All points a fixed distance from a point → a circle.
- All points equidistant from two points → the perpendicular bisector of the line joining them.
- All points a fixed distance from a line → two parallel lines, one each side, with rounded ends.
| The rule | The locus |
|---|---|
| Fixed distance from a point | A circle of that radius |
| Equidistant from two points | Perpendicular bisector of the join |
| Equidistant from two lines | Angle bisector |
| Fixed distance from a line | Two parallel lines, rounded ends |
In a nutshell: perpendicular bisector = equidistant from two points; angle bisector = equidistant from two lines; the locus a fixed distance from a point is a circle. Keep the compass fixed and leave the arcs in.