Constructions, loci, bearings and symmetry
Ruler-and-compass constructions, describing positions, and symmetry.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 4.4.1Carry out standard constructions and use loci to describe regions.
- 4.4.2Use three-figure bearings and describe symmetry properties.
6 minute read
Constructions, loci, bearings and symmetry
Constructions
All require compasses and a straight edge, and the arcs must be left visible — they are the evidence of method and carry the marks.
- Perpendicular bisector of AB: arcs from A and B of equal radius, above and below; join the crossings. Every point on it is equidistant from A and B.
- Angle bisector: arc from the vertex crossing both arms, then equal arcs from those two crossings; join to the vertex. Every point on it is equidistant from the two lines.
- Perpendicular from a point to a line: arc from the point crossing the line twice, then bisect that segment.
Loci
A locus is the set of all points satisfying a rule:
- Fixed distance from a point → a circle.
- Fixed distance from a line → parallel lines either side, with semicircular ends.
- Equidistant from two points → the perpendicular bisector.
- Equidistant from two lines → the angle bisector.
Region questions combine these: shade where all conditions overlap, and use a dashed boundary if the boundary itself is excluded.
Bearings
Three rules, all compulsory:
- Measured from north.
- Clockwise.
- Always three figures — so 45° is written 045°.
A back bearing (the return journey) differs by 180°: add 180 if the bearing is under 180, subtract 180 if it is over.
Symmetry
- Line symmetry — the number of mirror lines. A regular n-sided polygon has n.
- Rotational symmetry — the order is how many positions look identical in a full turn. A regular n-gon has order n.
- In 3D, shapes have planes of symmetry rather than lines.
Think of it like this
A locus is the trail left by every point obeying a rule — like the mark a goat on a fixed rope wears into a field. The rope length gives a circle; two ropes give the overlap.
Worked examples
Method, step by step
The bearing of B from A is 070°. Find the bearing of A from B.
- 1The back bearing differs by 180°.
- 2Since 070° is less than 180°, add: 070 + 180 = 250.
- 3Write it as three figures: 250°.
250°
Common misconceptions
- Rubbing out construction arcs to make the diagram tidy. The arcs are the working, and removing them removes the marks.
- Writing a bearing as 45° instead of 045°. Three figures are compulsory.
- Measuring a bearing anticlockwise or from south. Always clockwise from north.
In the exam
- Never measure a construction with a protractor when the question says "construct" — compass arcs must be visible.
- For a region question, shade the overlap and state clearly which conditions it satisfies.