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Geometry & Measures · Foundation & Higher

Constructions & loci

Constructions use a ruler and compasses to draw accurate geometric figures. Loci are sets of points satisfying a given condition — they are often arcs or straight lines.

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Key facts to remember

  • 1Perpendicular bisector of AB: the locus of points equidistant from A and B — drawn with compasses.
  • 2Angle bisector: the locus of points equidistant from two lines — drawn with compasses.
  • 3Locus of points a fixed distance r from a point = circle of radius r.
  • 4Locus of points equidistant from two parallel lines = parallel line halfway between them.
  • 5Always leave construction arcs visible — do not rub them out.
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Worked examples

Example 1

Describe the locus of points that are exactly 3 cm from a fixed point P.

Working

  1. The locus is all points at distance 3 cm from P
  2. This forms a circle with centre P and radius 3 cm
AnswerA circle with centre P and radius 3 cm
Example 2

Construct the perpendicular bisector of line segment AB of length 8 cm.

Working

  1. Open compasses to more than half of AB (e.g. 5 cm)
  2. Draw arcs above and below AB from point A
  3. Without changing compass width, draw arcs from B — they cross the previous arcs
  4. Join the two crossing points — this is the perpendicular bisector
AnswerA line through the midpoint of AB at 90° to AB
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Common mistakes

✗Rubbing out construction arcs — they must be left to show the method.
✗Changing compass width between the two sets of arcs when constructing bisectors.
✗Drawing loci as a single point rather than the full set of points.
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Exam tips

✓Use a sharp pencil and keep compass arcs light but visible — never use a ruler for curved loci.
✓Read loci questions carefully: "at least 3 cm" means shade the region outside a circle; "no more than 3 cm" means shade inside.

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