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Learn8300Papers 1–3 · Foundation tierGeometry and measuresConstructions and loci
Geometry and measures · Subtopic 5 of 14

Constructions and Loci

A locus is the set of all points that obey a rule. Points a fixed distance from a point make a circle; equidistant from two points, the perpendicular bisector; equidistant from two lines, the angle bisector. Ruler-and-compass constructions draw these exactly, and you leave the arcs to show your method.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 6 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • Construct a triangle from three sides with ruler and compasses.
  • Construct perpendicular bisectors and angle bisectors.
  • Describe and draw loci of points from a point, a line, two points and two lines.
  • Combine loci to find a region, using a scale drawing.
Pause and think

A goat is tied to a post by a 44 m rope in an open field. Describe the shape of grass it can reach.

Section 2 of 14

Core idea

Start hereWhere to put the mobile mast

A phone mast must be equally far from two towns and within 5 km of a road. Each rule traces a set of points, a locus; the answer is where the loci overlap. Compasses and a ruler are all you need.

The big idea

Fixed distance from a point: circle. Equidistant from two points: perpendicular bisector. Equidistant from two lines: angle bisector.

1. Constructions

Perpendicular bisector of AB with compass arcs
equidistant from A and BAB
Arcs of the same radius from A and B cross twice; join the crossings. Every point on that line is the same distance from A as from B.
Why the perpendicular bisector works
  1. PA=PB for each arc crossing PPA=PB\text{ for each arc crossing }P
    Equal compass radius from AA and BB.
  2. two such points determine the line\text{two such points determine the line}
    Every point on it is equidistant from AA and BB.
  • Perpendicular bisector: equal arcs from both ends, join crossings.
  • Angle bisector: arc from the vertex, then equal arcs from the two crossings, join.
  • Triangle from three sides: arcs from each end of the base.

2. Loci

A goat on a 3 m rope tied at a corner of a shed
shedgrazing region
The goat can reach three-quarters of a circle of radius 3 m; the shed blocks the rest. Loci questions are often about regions like this.
Worked exampleA goat on a rope

Tied with a 3 m rope at the outside corner of a square shed.

Goal
Grazing area.
Strategy
The shed blocks one quarter.
  1. 34 of a circle\tfrac34\text{ of a circle}
    270∘270^\circ is open.
  2. 34π(32)=21.2 m2\tfrac34\pi(3^2)=21.2\text{ m}^2
    Area.
Answer21.221.2 m² (if the shed sides exceed 3 m).
✓ CheckLess than the full circle 28.328.3 ✓.
Exam tipLeave arcs Construction arcs must be visible for the marks.
Pause and think

Locus of points 2 cm from a line segment?

Summary

  • Keep arcs visible.
  • Three basic loci.
Key words
Locus
Set of points obeying a rule.
HistoryEuclid's rules. In the Elements (about 300300 BC) Euclid allowed only two tools: a straight edge with no markings and a pair of compasses. Every construction you learn here is in Book I, written over 23002300 years ago.
Fun factA teenager's 17-gon. For 20002000 years nobody knew how to construct a regular 1717-sided polygon with ruler and compasses. In 17961796, aged 1919, Carl Friedrich Gauss showed how. He was so proud he asked for a 1717-gon on his gravestone (the stonemason said it would just look like a circle).
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