GCSE Maths · Loci, Bearings and Vectors · Foundation · Lesson 22.2

GCSE Maths: Ruler and compass constructions (Foundation)

Part of the GCSE Maths loci, bearings and vectors chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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22.2

Ruler and compass constructions

Edexcel 1MA1: G2
Grade 3-6

To construct something in geometry means to draw it accurately using only a ruler (for straight lines) and a pair of compasses (for arcs and circles). A protractor is used only to check your work. Constructions give exact results, because the compasses make every point on an arc exactly the same distance from the centre.

Rules for good constructions. Use a sharp pencil. Keep the compasses tight so the radius does not slip. Never rub out the construction arcs: they show that you used the correct method, and an examiner gives marks for them. When two arcs need the same radius, do not change the compasses in between.

1. The perpendicular bisector of a line AB. This is the line that cuts AB in half (bisects it) and meets it at 90° (is perpendicular to it). Every point on this line is the same distance from A and from B.

Open the compasses to more than half the length of AB. Put the point on A and draw an arc above and an arc below the line.
Keep the same radius. Put the point on B and draw two more arcs that cross the first two.
Use a ruler to join the two places where the arcs cross. This line is the perpendicular bisector. It cuts AB at its midpoint M.

Perpendicular bisector of AB. The four arcs all have the same radius, so P and Q are each the same distance from A and B. The line PQ cuts AB at its midpoint M and makes 90° with it.

Why does it work? The four arcs have equal radii, so the four points A, P, B, Q are joined by four equal lengths: AP = PB = BQ = QA. That shape is a rhombus, and the diagonals of a rhombus cross each other at 90° and cut each other in half.

2. The bisector of an angle. To bisect an angle means to cut it into two equal angles.

Put the compass point on the corner O. Draw an arc that crosses both arms. Call the crossing points P and Q.
Put the point on P and draw an arc inside the angle. Keep the same radius, put the point on Q and draw another arc so the two arcs cross at R.
Use a ruler to draw a straight line from O through R. This line is the angle bisector.

Bisecting an angle of 64°. OR cuts it into two angles of 32°. Every point on OR is the same distance from the two arms.

3. The perpendicular from a point P to a line. The perpendicular from P to the line is the shortest path from P to the line. This is the “perpendicular distance”.

Put the compass point on P and draw an arc that crosses the line twice. Call the crossing points X and Y.
Put the point on X and draw an arc on the other side of the line from P. Keep the same radius, put the point on Y and draw another arc to cross it at Z.
Join P to Z with a ruler. The line PZ crosses the line at F, and PF is perpendicular to the line.

The perpendicular from P to the line. The length PF is the shortest distance from P to the line.

4. The perpendicular at a point P on a line. This makes a right angle at P.

Put the compass point on P and draw arcs that cross the line on each side of P, at X and Y. (P is exactly in the middle of X and Y.)
Open the compasses wider. Put the point on X and draw an arc above the line. Keep the same radius, put the point on Y and draw an arc to cross it at Z.
Join P to Z. The line PZ is perpendicular to the line at P.

A right angle at P on the line. Notice that this is the same idea as the perpendicular bisector of XY.

5. An angle of 60°. Draw a line AB. Put the compass point on A and draw an arc that crosses the line at P. Keep the same radius, put the point on P and draw an arc that crosses the first arc at Q. Join A to Q. The triangle APQ has three equal sides, so it is equilateral and angle QAP is 60°. From 60° you can make 30° by bisecting, and 120° by constructing two 60° angles side by side.

Constructing 60°. AP = AQ = PQ (the same radius each time), so angle QAP is 60°.

Key ideas

Perpendicular bisector of AB: equal arcs from A and from B, join the crossing points. It cuts AB in half at 90°.
Angle bisector: arc from the corner, then equal arcs from the two crossing points, join the corner to where those arcs cross.
Perpendicular from a point to a line: arcs from the point cut the line twice, then equal arcs from those two points. This gives the shortest distance.
60° is made from an equilateral triangle: use the same radius for both arcs.
Never rub out your construction arcs.

Worked example 22.2.1

Draw a line AB that is 8 cm long. Construct the perpendicular bisector of AB.

Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.

Method. Use equal arcs from A and from B (more than 4 cm), then join the two places where the arcs cross.

Draw AB with a ruler. The line must be exactly 8 cm long because we are told its length.
AB = 8 cm
Draw arcs from A. The radius must be more than half of AB (more than 4 cm), or the arcs from A and B will not meet. Use 5.2 cm.
Compass point on A: draw one arc above AB and one below it.
Draw arcs from B. Use the same compass setting so that the two points where the arcs cross are the same distance from A and from B.
Compass point on B: draw arcs that cross the first two at P and Q.
Join P and Q. The line PQ is the perpendicular bisector. Leave all the arcs.

Check. Measure: the line crosses AB at M with AM = MB = 4 cm and the angle at M is 90°. ✓

Watch out. If the compass setting is half of AB or less, the arcs will not cross. Always open the compasses wider than half the line.

Worked example 22.2.2

Draw an angle of 72°. Construct the bisector of the angle. What size is each of the two new angles?

Method. Draw the angle with a protractor, then use arcs from the corner and then from the two crossing points.

Draw the 72° angle. Use a ruler and a protractor for the angle you are given.
Corner O, arms OA and OB
Draw an arc from O. The arc must cross both arms. Call the crossing points P and Q.
Compass point on O; arc crosses OA at P and OB at Q.
Draw arcs from P and Q. Use the same radius for both. The arcs cross at R.
Compass point on P, then on Q, same setting: arcs cross at R.
Join O to R. OR is the angle bisector.
Work out each angle. A bisector cuts the angle into two equal halves.
72° ÷ 2 = 36° each

Check. Measure both new angles with a protractor: each should be 36°. ✓

Watch out. Use the compasses for the construction. Marking the half-angle with a protractor loses the marks for the method.

Worked example 22.2.3

The diagram shows a point P that is 2.8 cm above a straight line. Construct the perpendicular from P to the line. How far is P from the line?

Method. Use arcs from P to find X and Y, then equal arcs from X and Y to find Z. Join P to Z.

Draw an arc from P. Open the compasses wide enough to cross the line twice. Draw an arc crossing the line at X and Y.
Compass point on P.
Draw arcs from X and Y. On the far side of the line from P, draw arcs of the same radius from X and from Y. They cross at Z.
Radius more than half of XY.
Join P to Z. The line PZ meets the line at F, and PF is perpendicular to the line.
Measure PF. The perpendicular distance from P to the line is the shortest distance from P to the line.
PF = 2.8 cm

Check. Pick any other point on the line, such as X. The distance PX is the hypotenuse of a right-angled triangle with side PF, so PX is longer than PF. ✓

Watch out. The shortest distance from a point to a line is always the perpendicular distance, never a slanted one.

Worked example 22.2.4

Construct an angle of 30° using only a ruler and compasses.

Method. Build 30° from the angles you can construct exactly: 60° (arc method), 90° (perpendicular) and halves of them (bisector).

Draw a straight line and mark the point A. The angle will have its corner at A, with one arm along the line.
Use a ruler. Make the line about 8 cm long.
Construct 60°, then bisect it. 30° is half of 60°, so first make 60° and then use the angle bisector construction.
Keep the compasses at the same setting while you work, and keep all your arcs.
Join A to the new point. Draw the second arm of the angle with a ruler, passing through the intersection of the arcs.
Check with a protractor. A construction is accurate to about 1°. Measure the angle.
The angle should measure 30°.

Check. The protractor reading should be within 1° of 30°. ✓

Watch out. Use the compasses for the construction. A protractor is only for checking afterwards.

Worked example 22.2.5

A point P is marked on a straight line. Construct a line that is perpendicular to the line at P.

Method. Use arcs from P to find X and Y, then equal larger arcs from X and Y.

Mark X and Y. Compass point on P: draw arcs that cross the line on both sides of P. P is now exactly half way between X and Y.
PX = PY
Draw arcs from X and Y. Open the compasses wider (more than PX). Keep the setting the same for both arcs. They cross at Z.
Compass point on X, then on Y: arcs cross at Z above the line.
Join P to Z. Use a ruler. Leave the arcs.

Check. Place a set square or protractor on P: the angle between PZ and the line should be 90°. ✓

Watch out. X and Y must be the same distance from P. Without that, the line is not perpendicular.

Watch out

Do not change the compass setting between the two arcs that must cross. If the radius changes the crossing point is not the same distance from both centres and the line will not be a true bisector. Do not rub out arcs.

Exercise 22C Constructions: method and accuracy

Non-calculator

FPut the steps in the correct order. Write the letters.

(a)Constructing the perpendicular bisector of AB:
A Keep the same radius, put the point on B and draw arcs that cross the first two.
B Draw the line AB with a ruler.
C Join the two crossing points with a ruler.
D Open the compasses to more than half the length of AB.
E Put the compass point on A and draw an arc above and below AB.

(b)Bisecting an angle:
A Keep the same radius and draw an arc from the second crossing so that the arcs cross.
B Put the compass point on the corner O and draw an arc that crosses both arms.
C Draw a straight line from O through the point where the arcs cross.
D Put the point on the first crossing and draw an arc inside the angle.

Done

FFill in each gap.

(a)The perpendicular bisector of AB meets AB at an angle of ___ degrees.

(b)The perpendicular bisector of AB cuts AB into two ___ parts.

(c)When you bisect an angle, the first arc is drawn with the compass point on the ___ of the angle.

(d)You must never ___ out the construction arcs.

Done

FDraw a line AB that is 8 cm long. Construct the perpendicular bisector of AB. Mark the midpoint M. What is the length AM?

Done

FConstruct the bisector of each angle. State the size of each of the two equal angles.

(a)An angle of 50°

(b)An angle of 80°

(c)An angle of 120°

(d)An angle of 140°

Done

FDescribe how to construct each angle using only a ruler and compasses. What does the angle measure when you check it with a protractor?

(a)An angle of 60°

(b)An angle of 120°

(c)An angle of 45°

(d)An angle of 90°

Done

FEach diagram is drawn actual size. Construct the perpendicular from P to the line, then measure the shortest distance from P to the line.

(a)

(b)

Done

RAisha rubs out her construction arcs after she has drawn a perpendicular bisector. She says: “The line is right, so the arcs are not needed.” Explain why she should leave the arcs on her work.

Done

PSA straight path runs along a line. A point P is on the path. Construct the perpendicular to the path at P. Q is the point on this perpendicular that is 4 cm from P. The point A is on the path 3 cm from P. Work out the distance AQ.

Done

RExplain why the perpendicular distance from a point P to a line is shorter than the distance from P to any other point on the line. Use P that is 3 cm from the line and a point X on the line that is 4 cm along from the foot of the perpendicular.

Done