A locus is the set of all points that satisfy a rule. Points a fixed distance from a point lie on a circle; points equidistant from two points lie on the perpendicular bisector; and points equidistant from two lines lie on the angle bisector. Constructions draw these accurately with a ruler and compasses, and combined conditions define a region.
Key loci
- Fixed distance from a point: a circle
- Fixed distance from a line segment: two parallel lines joined by semicircles
- Equidistant from two points: the perpendicular bisector
- Equidistant from two lines: the angle bisector
- Leave construction arcs visible
The two key constructions
The perpendicular bisector of AB is drawn with equal arcs from A and from B, crossing above and below the line; joining the crossing points gives a line that cuts AB in half at 90°. The angle bisector is drawn by marking equal arcs on both arms from the vertex, then equal arcs from those marks. Keep the compass setting fixed for each pair of arcs, and leave all arcs showing.
Distance from a point or a line
All the points a fixed distance r from a point P form a circle of radius r, centre P. All the points within r of P form the region inside it. All the points a fixed distance from a straight line segment form a running-track shape: two parallel lines joined by a semicircle at each end.
Equidistant from two points or two lines
Points the same distance from two fixed points A and B lie on the perpendicular bisector of AB. Points the same distance from two intersecting lines lie on the bisector of the angle between them. “Closer to A than to B” means the side of the perpendicular bisector that contains A.
Grazing problems
An animal tied to a post by a rope can reach a circle with the rope as radius. A goat on an 8 m rope in an open field can graze π × 8² = 201.1 m². If the post is at the corner of a barn and the barn blocks a quarter of the circle, it can graze three quarters: 150.8 m².
Regions from several conditions
Many questions combine conditions, such as “within 5 m of P and closer to A than to B”. Draw each locus, then shade only the region that satisfies every condition. A region between two circles of radii 2 cm and 6 cm, centre O, has area π(6² − 2²) = 32π = 100.5 cm².
Worked examples
Example 1. Describe fully the locus of points that are the same distance from A as from B.
- Equidistant from two points means the perpendicular bisector.
Answer: The perpendicular bisector of the line AB.
Example 2. Describe the locus of points that are exactly 3 cm from a point P.
- Every such point is 3 cm from P in some direction.
Answer: A circle of radius 3 cm with centre P.
Example 3. A goat is tied to a post in an open field with an 8 m rope. Work out the area it can graze, to 1 decimal place.
- The goat can reach a circle of radius 8 m.
- π × 8² = 64π = 201.06…
Answer: 201.1 m².
Example 4. The post is moved to the corner of a barn, which blocks a quarter of the circle. Work out the new grazing area, to 1 decimal place.
- The goat can reach three quarters of the circle.
- ¾ × 64π = 48π = 150.79…
Answer: 150.8 m².
Example 5. A region lies at least 2 cm and at most 6 cm from a point O. Work out its area, to 1 decimal place.
- Area between two circles: π × 6² − π × 2².
- 36π − 4π = 32π = 100.53…
Answer: 100.5 cm².
Common mistakes
- Rubbing out construction arcs. Leave all arcs visible; they earn the method marks.
- Changing the compass setting mid-construction. Keep the same radius for each matching pair of arcs.
- Confusing the two bisectors. Equidistant from points: perpendicular bisector. From lines: angle bisector.
- Drawing a rectangle for distance from a segment. The ends are semicircles, not square corners.
- Shading the wrong side. Test a point to check it satisfies every condition.
How loci and constructions are examined
Loci and construction questions need a ruler, compasses and careful drawing, and they appear on Foundation and Higher papers for GCSE and iGCSE. Questions ask you to:
- construct a perpendicular bisector or an angle bisector
- draw the locus of points a given distance from a point or line
- shade a region that satisfies several conditions
- solve a context problem, such as where a phone mast or a goat can reach
Practise with the loci and construction worksheet, which includes answers, and the geometry revision engine.
Grazing areas use circle and sector area, and loci use the facts in angle rules.
Frequently asked questions
What is a locus?
The set of all points that follow a given rule, such as being 4 cm from a point.
What is the locus of points equidistant from two points?
The perpendicular bisector of the line joining them.
Why must construction arcs be left on the diagram?
They show the method, which earns marks.