Ruler-and-compass constructions
GCSE expects you to perform constructions accurately, leaving construction arcs as evidence. The standard set you must know:
- Perpendicular bisector of a line segment.
- Angle bisector.
- Perpendicular from a point to a line.
- Perpendicular from a point ON a line.
- Construct an equilateral triangle / 60° angle.
- Construct a triangle from given measurements (SSS, SAS, ASA).
Perpendicular bisector of AB
- Open compasses to MORE than half the length of AB.
- With compass on A, draw arcs above and below AB.
- Without changing the radius, repeat with compass on B.
- The arcs cross at two points; join them with a straight line.
This line is the perpendicular bisector — it cuts AB at right angles, exactly halfway.
Angle bisector at vertex B
- With compass on B, draw an arc cutting both rays of the angle (call points P and Q).
- From P, draw an arc inside the angle. From Q, draw an arc with same radius. They cross at R.
- Draw line BR — it bisects the angle.
Perpendicular from a point to a line
To drop a perpendicular from point P (not on the line) to a line:
- With compass on P, draw an arc cutting the line at A and B.
- Construct the perpendicular bisector of AB — it passes through P and is perpendicular to the line.
Constructing 60°
Draw a line; mark point A. Compass on A: draw arc cutting line at B. Without change, compass on B: arc cutting first arc at C. ∠CAB = 60°.
Constructing a triangle
For SSS (three sides):
- Draw the longest side as a base.
- From one end, set compass to the second side and draw an arc.
- From the other end, set compass to the third side and draw an arc.
- The intersection is the third vertex.
⚠Common mistakes
- Erasing construction arcs — examiners want to see them.
- Changing compass width between paired arcs (must be the same).
- Setting compass too small for a perpendicular bisector — must exceed half the segment.
- Drawing freehand instead of with a ruler.
- Confusing perpendicular bisector with angle bisector — different methods, different results.
➜Try this— Quick check
What angle is constructed by combining a 60° equilateral construction with an angle bisector?
- Bisecting 60° gives 30°.
AI-generated · claude-opus-4-7 · v3-deep-geometry