Perpendicular Bisector and Angle Bisector –

Compass-and-ruler constructions are a classic GCSE Maths topic that appear on both Foundation and Higher papers across AQA, Edexcel, and OCR. You must be able to construct a perpendicular bisector of a line segment and an angle bisector accurately, leaving all construction arcs visible. This guide gives you the exact steps for each construction along with exam tips and practice questions.

What Are Perpendicular Bisectors and Angle Bisectors?

A perpendicular bisector of a line segment is a line that cuts the segment exactly in half at right angles (90°). Every point on the perpendicular bisector is equidistant from the two endpoints of the segment.

An angle bisector is a line that divides an angle exactly in half. Every point on the angle bisector is equidistant from the two arms of the angle.

Key Formulas

Step-by-Step Method

Constructing a Perpendicular Bisector

1. Open your compass to more than half the length of the line segment.
2. Place the compass point on one end of the line segment and draw arcs above and below the line.
3. Without changing the compass width, place the point on the other end and draw arcs above and below, crossing the first pair of arcs.
4. Draw a straight line through the two points where the arcs intersect. This is the perpendicular bisector.
5. Leave all construction arcs visible — the examiner needs to see them.

Constructing an Angle Bisector

1. Place the compass point on the vertex of the angle and draw an arc that crosses both arms of the angle.
2. Place the compass on one intersection point and draw an arc between the two arms.
3. Without changing the compass width, place the compass on the other intersection point and draw another arc that crosses the previous one.
4. Draw a straight line from the vertex through the point where the two arcs cross. This is the angle bisector.
5. Leave all construction arcs visible.

Worked Example 1 — Foundation Level

Question: Construct the perpendicular bisector of a line segment AB that is 8 cm long.

Working:

Draw AB = 8 cm. Set compass to about 5 cm (more than half of 8 cm). Place compass on A and draw arcs above and below. Keep the same compass width, place on B, and draw arcs above and below. Mark the two intersection points and draw a straight line through them. This line passes through the midpoint of AB (4 cm from each end) at 90°.

Answer: The perpendicular bisector passes through the midpoint of AB at 90°.

Worked Example 2 — Higher Level

Question: Construct the bisector of an angle of 70°.

Working:

Draw the angle of 70° using a protractor. Place compass on the vertex and draw an arc crossing both arms. Place compass on each crossing point in turn and draw arcs between the arms. The two arcs intersect — draw a line from the vertex through this intersection. Each half of the angle is 35°.

Answer: Each half of the bisected angle is 35°.

Worked Example 3 — Exam Style

Question: A and B are two radio masts. A listener is equidistant from both masts. Construct the locus of all points equidistant from A and B.

Working:

The locus of points equidistant from A and B is the perpendicular bisector of AB. Construct the perpendicular bisector using the method above, leaving all arcs visible.

Answer: The locus is the perpendicular bisector of the line segment AB.

Common Mistakes

• Rubbing out construction arcs. The examiner awards marks for seeing the arcs. Never erase them.
• Setting the compass too small. If the compass width is less than half the line segment, the arcs will not cross. Always use a width greater than half the length.
• Moving the compass width between steps. The accuracy of the construction depends on keeping the compass width consistent within each set of arcs. If it slips, the result will be inaccurate.

Exam Tips

• Use a sharp pencil for accuracy — thick lines lose precision and may cost marks.
• Practise these constructions at home before the exam so the steps become automatic.
• The examiner may check accuracy to within 1 mm and 1° — take your time.
• Constructions are often part of loci questions, so mastering these two techniques is essential for the next step.
• Always label the key points and lines in your construction.

Practice Questions

Q1 (Foundation): Construct the perpendicular bisector of a line segment PQ of length 10 cm. Mark the midpoint M.

Q2 (Foundation): Construct the bisector of an angle of 80°. Measure each half to verify.

Q3 (Higher): Two points X and Y are 7 cm apart. Shade the region that is closer to X than to Y. Describe your method.

Practise constructions with instant feedback free on GCSEMathsAI.

Related Topics

• Loci and Regions
• Constructions and Loci
• Bearings

Summary

• A perpendicular bisector cuts a line segment in half at 90° and is constructed using two sets of compass arcs from each endpoint. An angle bisector divides an angle into two equal parts and is constructed using arcs from the vertex and the two points where the initial arc crosses the arms. Always leave construction arcs visible, use a compass width greater than half the segment, and keep a sharp pencil for accuracy.