Constructions and loci

With a ruler and compasses you can draw exact triangles, perpendiculars and bisectors, and use them to find regions that follow a rule.

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Triangles

Three sides (SSS): draw one side with a ruler. Set the compasses to the second length and draw an arc from one end; set them to the third length and draw an arc from the other end. Join to where the arcs cross.
Two sides and an angle or two angles and a side: use a ruler and protractor.
A triangle can only be drawn if the two shorter sides add up to more than the longest.

Perpendiculars

Perpendicular bisector of AB: equal arcs from A and B (more than half of AB), join the two crossing points.
Perpendicular from a point P to a line: an arc from P cuts the line twice; bisect between those two points.
Perpendicular at a point P on a line: mark equal distances either side of P, then bisect.
The perpendicular from a point to a line is the shortest distance to the line.
A B perpendicular bisector Keep the construction arcs showing.
Perpendicular bisector of AB: equal arcs from A and B, then join the crossing points.

Loci and regions

Fixed distance from a point: a circle.
Fixed distance from a line segment: two parallel lines joined by semicircles.
Equal distance from two points: the perpendicular bisector.
Equal distance from two lines: the angle bisector.
'Closer to A than to B' is the side of the perpendicular bisector containing A. Leave all arcs showing.
Worked example

A garden is drawn with scale 1 : 200. A path is 3.5 cm long on the drawing. How long is the real path?

  1. Real length = 3.5 × 200 = 700 cm
  2. 700 cm = 7 m

Answer: 7 m

Key idea

Use compasses for SSS triangles and all bisectors. The perpendicular from a point is the shortest distance to a line. Circles, perpendicular bisectors and angle bisectors are the three main loci.

Key words
Arc
Part of the circumference of a circle.
Segment
The region of a circle between a chord and an arc.
Parallel lines
Lines that never meet. They have the same gradient.

All GCSE and International GCSE Mathematics key terms and definitions →

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Extra practice

2 more questions

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  1. A boat is at B and the coast is a straight line. Which is the shortest route to the coast?

    1. The route to the nearest town
    2. The route that meets the coast at right angles
    3. The route that meets the coast at 45°
    4. Any straight route
    Show the answer

    Answer: The route that meets the coast at right angles

    The shortest distance from a point to a line is the perpendicular distance.

  2. A fountain must be the same distance from two straight walls that meet at a corner. Where can it go?

    1. On the perpendicular bisector
    2. On one of the walls
    3. On the bisector of the angle between the walls
    4. On a circle around the corner
    Show the answer

    Answer: On the bisector of the angle between the walls

    Points equidistant from two lines lie on the angle bisector.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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