Constructions with Ruler and Compasses

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

A construction is an accurate drawing made with a ruler, a pair of compasses and (for triangles with given angles) a protractor. The arcs you draw with the compasses are your working — in a test you must leave them on the page so the examiner can see your method. On this page you will learn the three ways to construct a triangle, how to find the exact middle of a line and draw a line at right angles through it, and how to cut an angle exactly in half.

1. Key words and equipment

• Use a sharp pencil — a thick line can be 1 mm wide, which is already an error.
• Use stiff compasses that do not slip, and keep the point firmly on the centre.
• Measure lengths from the 0 mark of the ruler, not from the end of the ruler.
• Never rub out your arcs. They are the evidence of the construction.

2. Constructing triangles

You can construct a triangle when you know enough about it. There are three common cases:

Case 1: three sides (SSS) — Construct triangle ABC with AB = 7 cm, AC = 5 cm and BC = 6 cm.

1. Draw the longest side, AB = 7 cm, with a ruler. Label A and B.
2. Open the compasses to 5 cm (measure on the ruler). Put the point on A and draw an arc above AB.
3. Open the compasses to 6 cm. Put the point on B and draw an arc that crosses the first arc.
4. The crossing point is C. Join AC and BC with a ruler.
5. Leave the arcs. Label the lengths.

Case 2: two sides and the included angle (SAS) — Construct triangle ABC with AB = 6 cm, angle BAC = 50° and AC = 4.5 cm.

1. Draw AB = 6 cm.
2. Put the centre of the protractor on A, with the base line along AB. Mark 50° (use the scale that starts at 0 on AB).
3. Draw a line from A through your mark.
4. Measure 4.5 cm along this line from A and mark C.
5. Join BC. If asked, measure BC: it is about 4.6 cm.

Case 3: two angles and the included side (ASA) — Construct triangle ABC with AB = 7 cm, angle A = 40° and angle B = 65°.

1. Draw AB = 7 cm.
2. At A, measure 40° with the protractor and draw a long line.
3. At B, measure 65° (measured from BA) and draw a long line.
4. The lines cross at C. Check: angle C should be 180 − 40 − 65 = 75°.

3. The perpendicular bisector and the midpoint

The perpendicular bisector of a line segment AB cuts AB into two equal halves and crosses it at 90°. The point where it crosses AB is the midpoint M. You construct it with compasses only — no protractor and no measuring of the halves.

1. Open the compasses to more than half the length of AB.
2. Put the point on A. Draw an arc above AB and an arc below AB.
3. Without changing the compasses, put the point on B and draw arcs that cross the first two arcs.
4. Use a ruler to draw a straight line through the two crossing points. This is the perpendicular bisector.
5. Where it crosses AB is the midpoint M. (Check: AM = MB and the angle is 90°.)

Why does it work? Each crossing point is the same distance from A as from B (both arcs have the same radius). Every point on the perpendicular bisector is equidistant (the same distance) from A and from B. This is useful in real problems: 'a tree is planted the same distance from two gates' means it lies on the perpendicular bisector of the line joining the gates.

4. The angle bisector

The angle bisector cuts an angle into two equal angles. Again you use compasses, not a protractor.

1. Put the compass point on the vertex O. Draw an arc that crosses both arms of the angle. Label the crossing points P and Q.
2. Put the compass point on P and draw an arc inside the angle.
3. Keep the same radius, put the point on Q and draw an arc that crosses the last arc at R.
4. Draw a straight line from O through R. This is the angle bisector.

Why does it work? OP = OQ (the same arc) and PR = QR (the same compass setting), and OR is shared. So triangles OPR and OQR have three equal sides — they are identical (congruent) — which means angle POR = angle QOR.

5. Special angles with compasses only

• 60°: draw a line OP. With the compasses set to OP, draw an arc from O and an arc from P; they cross at Q. Triangle OPQ is equilateral, so angle QOP = 60°.
• 30°: bisect a 60° angle.
• 90°: draw the perpendicular bisector of a line segment (or bisect a straight angle of 180°).
• 45°: bisect a 90° angle.

6. Worked examples

Worked example 1 — Construct triangle PQR with PQ = 8 cm, PR = 6 cm and QR = 5 cm. Leave your construction arcs. [2]

1. Draw PQ = 8 cm (the longest side) as the base.
2. Compasses at 6 cm, point on P: arc above PQ.
3. Compasses at 5 cm, point on Q: arc crossing the first arc. Label the crossing R.
4. Join PR and QR. Check by measuring: PR = 6 cm and QR = 5 cm.
5. Aim for every side to be correct to within 1 mm, and keep both arcs visible as evidence of the construction.

Common mistake: drawing PR and QR with a ruler by 'trying' different positions. Without arcs there is no evidence of construction.

Worked example 2 — Construct triangle ABC with AB = 6 cm, angle BAC = 50° and AC = 4.5 cm. Measure BC. [3]

1. Follow Case 2 above.
2. Measure BC with a ruler: BC ≈ 4.6 cm (an answer between 4.5 cm and 4.7 cm shows an accurate drawing).

Worked example 3 — Construct triangle XYZ with XY = 7 cm, angle X = 40° and angle Y = 65°. Measure XZ. [3]

1. Draw XY = 7 cm; draw 40° at X and 65° at Y; the lines meet at Z.
2. Measure XZ: about 6.6 cm. Check angle Z with the protractor: it should be 75°.

Worked example 4 — Draw a line AB = 9 cm. Construct the perpendicular bisector of AB and mark the midpoint M. How far is M from A? [2]

1. Set the compasses to more than 4.5 cm (for example 6 cm). Draw arcs above and below AB from A, then from B.
2. Join the crossing points. Mark M where this line crosses AB.
3. AM = 9 ÷ 2 = 4.5 cm. Check with the ruler.

Common mistake: setting the compasses to less than half of AB. The arcs from A and B then do not meet.

Worked example 5 — Draw an angle of 110°. Construct its bisector. What is the size of each new angle? [2]

1. Draw the 110° angle with a protractor and label the vertex O.
2. Arc from O crossing both arms at P and Q; equal arcs from P and Q meeting at R; join OR.
3. Each angle = 110 ÷ 2 = 55°. Measure both to check.

Worked example 6 (thinking mathematically) — Amir wants to construct a triangle with sides 4 cm, 5 cm and 10 cm. Explain why he cannot. [1]

1. The two shorter sides add up to 4 + 5 = 9 cm, which is less than 10 cm.
2. The arcs of radius 4 cm and 5 cm from the ends of the 10 cm line cannot reach each other, so they never cross — no triangle can be made.

Practice

All pages in this course

Start here

• Stage 8 Maths: The Tests and How to Prepare

Number

• Order of Operations and Calculating with Integers
• Factors, Multiples, Primes, HCF and LCM
• Types of Number and Index Laws
• Squares, Cubes and Roots (Positive and Negative)
• Multiplying and Dividing by 0.1 and 0.01, and Significant Figures
• Fractions: Recurring Decimals, Mixed Numbers and Order of Operations
• Percentage Increase, Decrease and Absolute Change
• Ordering and Calculating with Decimals
• Ratio and Proportion

Algebra

• Algebraic Expressions: Simplifying, Expanding and Factorising
• Writing Expressions and Formulae, and Changing the Subject
• Equations and Inequalities
• Sequences and the nth Term
• Functions and Straight-line Graphs (y = mx + c)
• Reading and Interpreting Real-life Graphs

Geometry and measure

• Quadrilaterals and Regular Polygons
• Circumference of a Circle, and Miles and Kilometres
• Area of Parallelograms and Trapezia
• 3D Shapes: Euler's Formula and Plans and Elevations
• Volume of Prisms and Surface Area
• Angles in Triangles and on Parallel Lines
• Constructions with Ruler and Compasses
• Bearings and Midpoints
• Translation, Reflection, Rotation and Enlargement

Statistics and probability

• Data Collection and Sampling
• Representing Data
• Averages, Range and Comparing Distributions
• Probability: Complementary and Combined Events

Practice tests

• Stage 8 Maths — Practice Test 1
• Stage 8 Maths — Practice Test 2
• Stage 8 Maths — Practice Test 3