15.6 Constructions

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: perpendicular bisectors, angle bisectors, perpendiculars and constructing 60°, 90°, 30° and 45°. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Construct the perpendicular bisector of a line segment
• Construct the bisector of an angle
• Construct a perpendicular from a point to a line, and at a point on a line
• Construct angles of 60°, 30°, 90° and 45° using only a ruler and compasses

Key words

• Bisect: Cut exactly in half.
• Perpendicular: At right angles (90°) to a line.
• Perpendicular bisector: The line that cuts a line segment in half at right angles.
• Angle bisector: The line that cuts an angle in half.
• Equidistant: The same distance from two or more points or lines.

Perpendicular bisector of a line

Worked example: bisect a 6 cm line AB

1. Open the compasses to more than half of AB, say 4 cm.
2. With the point on A, draw arcs above and below the line.
3. Keep the same radius. With the point on B, draw arcs that cross the first two.
4. Join the two crossing points with a ruler.

Answer: the line crosses AB at its midpoint (3 cm from A) at 90°

Key fact: Every point on the perpendicular bisector of AB is equidistant from A and B.

Bisecting an angle

Worked example: bisect an angle of 64°

1. With the point on O, draw an arc crossing both arms.
2. From each crossing point draw an arc (same radius) inside the angle so the two arcs cross.
3. Join O to the point where the arcs cross.

Answer: two angles of 32°

Every point on the angle bisector is equidistant from the two arms of the angle.

Perpendicular from a point to a line

Worked example: perpendicular from P

1. With the point on P, draw an arc that crosses the line twice.
2. From each of these two points, draw arcs of equal radius below the line so they cross.
3. Join P to the crossing point. The line meets the original line at 90°.

Answer: the perpendicular from P, which is the shortest distance from P to the line

For a perpendicular at a point Q on a line, draw arcs of equal radius either side of Q on the line, then construct the perpendicular bisector of those two points.

Constructing 60°, 30°, 90° and 45°

Worked example: an angle of 60° at A

1. With the point on A, draw a large arc crossing the line at B.
2. With the same radius and the point on B, draw an arc crossing the first arc at C.
3. Join A to C. AB = AC = BC, so triangle ABC is equilateral.

Answer: angle CAB = 60°

• 60°; How to construct it equilateral triangle construction
• 30°; How to construct it construct 60°, then bisect it
• 90°; How to construct it perpendicular at a point on a line
• 45°; How to construct it construct 90°, then bisect it

Common mistakes

• Changing the compass width halfway through a perpendicular bisector. Both pairs of arcs must use the same radius.
• Opening the compasses to less than half the line, so the arcs never cross.
• Rubbing out the arcs. They are the evidence that you constructed the line.
• Using a protractor when the question says "use ruler and compasses only".

Revision questions

Try these, then check your answers below each one.

Question 1. An angle of 84° is bisected. What size is each new angle, in degrees?

Answer: 42 °

Working: 84 ÷ 2 = 42°.

Question 2. Which construction gives all the points that are the same distance from A and from B? (Choose from: The perpendicular bisector of AB, The angle bisector at A, A circle centred on A, A line parallel to AB.)

Answer: The perpendicular bisector of AB

Working: Every point on the perpendicular bisector of AB is equidistant from A and B.

Question 3. AB is 9 cm long. Its perpendicular bisector crosses AB at M. How far is M from A, in cm?

Answer: 4.5 cm

Working: 9 ÷ 2 = 4.5 cm.

Related lessons

• Previous lesson: 15.5 Accurate Drawings 2
• Next lesson: 15.7 Loci and Regions
• In this chapter: 15.1 3D Solids
• In this chapter: 15.2 Plans and Elevations
• In this chapter: 15.3 Accurate Drawings 1

Continue in the full course

This lesson is section 15.6 (EGCMF1506) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.