EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Constructions 1
Transformations and constructions · Lesson 6 of 8
Warm-up
Answer each one, then check.
1. What do the angles in a triangle add up to?
180°
2. What is a perpendicular line?
One that meets another at 90°
3. What does bisect mean?
Cut exactly in half
4. What is the longest side of a right-angled triangle called?
The hypotenuse
5. What is the midpoint of a line 8 cm long?
The point 4 cm from each end
Learning Objectives
1. Construct a triangle given three sides (SSS).
2. Construct a triangle given two sides and the angle between them (SAS).
3. Construct the perpendicular bisector of a line segment.
4. Explain why a triangle can or cannot be constructed.
The Rules of Construction
Use only a pencil, a ruler for straight lines, and compasses.
▸ Keep the construction arcs. Do not rub them out: they show your method and earn marks.
▸ Sharp pencil. Thin lines make the intersections easy to see.
▸ Keep the compasses fixed. Do not change the radius between two arcs that must be equal.
▸ Label the lengths. Write the length beside each side, and mark any angles.
Constructing a Triangle from Three Sides
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The point where the two arcs cross is the third vertex. |
Triangle ABC constructed with a base of 6 centimetres and two arcs of radius 5 and 4 centimetres meeting above it.
Constructing an SSS Triangle
For a triangle with sides 6 cm, 5 cm and 4 cm.
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1 Draw the longest side Measure 6 cm with a ruler and label the ends A and B. |
2 Set the compasses to 5 cm Put the point at A and draw an arc above the line. |
3 Set the compasses to 4 cm Put the point at B and draw a second arc crossing the first. |
4 Mark C where the arcs cross Join C to A and C to B with a ruler. |
5 Label and check Measure to check each side and keep the arcs. |
Can the Triangle Be Made?
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Can a triangle be constructed with sides 3 cm, 4 cm and 8 cm? |
1. The two shorter sides must add up to more than the longest
3 + 4 = 7
2. Compare with the longest side
7 < 8
3. The arcs would not meet
The two short sides cannot reach each other
Answer: No: 3 + 4 < 8, so the arcs do not cross and no triangle exists.
An SAS Triangle
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Construct a triangle with sides of 5 cm and 7 cm and an angle of 40° between them. |
1. Draw the 7 cm side
Label the ends A and B
2. Measure 40° at A
Use a protractor and mark the angle
3. Draw a line from A through the mark
Measure 5 cm along it and label C
4. Join B to C
The triangle ABC is complete
Answer: Triangle ABC with AB = 7 cm, AC = 5 cm and angle A = 40°.
The Perpendicular Bisector
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Every point on the perpendicular bisector is the same distance from A and B. |
A line segment AB with two pairs of equal arcs crossing above and below it and a line through the crossings meeting AB at a right angle at its midpoint.
Constructing the Perpendicular Bisector
The same radius from both ends.
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1 Open the compasses more than half of AB Any radius bigger than half will do |
2 Draw arcs from A above and below the line Keep the same radius |
3 Draw arcs from B, crossing the first arcs Do not change the radius |
4 Mark the two crossing points One above and one below AB |
5 Join the crossing points with a ruler This line bisects AB at 90° |
Which Construction?
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SSS TRIANGLE |
SAS TRIANGLE |
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▸ You know all three sides. ▸ Use two arcs to find the third vertex. ▸ Always check the two shorter sides add to more than the longest. |
▸ You know two sides and the angle between them. ▸ Use a protractor for the angle. ▸ The third side is found by joining the ends. |
Key Terms
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Construct Draw accurately using only a ruler and compasses (a protractor when an angle is given). |
Compasses An instrument for drawing arcs and circles. |
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Arc Part of the circumference of a circle. |
Bisect To cut exactly in half. |
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Perpendicular bisector A line at 90° through the midpoint of a segment. |
Vertex A corner of a shape. |
Your Task: Construct and Measure
15 minutes
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Construct a triangle with sides 7 cm, 6 cm and 5 cm. Measure the three angles with a protractor and add them up. Then construct the perpendicular bisector of the 7 cm side and check it passes through the opposite vertex or not. 1. Construct with compasses. 2. Measure the angles. 3. Check the sum. |
A good answer shows: The angles are about 78°, 57° and 44° (sum 180°). The perpendicular bisector of the 7 cm side does not pass through the opposite vertex because the triangle is scalene; it would only do that for an isosceles triangle with equal sides on either side.
Can I...?
☐ Draw a line of exact length.
☐ Construct an SSS triangle.
☐ Construct an SAS triangle.
☐ Use a protractor accurately.
☐ Construct a perpendicular bisector.
☐ Keep construction arcs.
☐ Test whether three lengths make a triangle.
☐ Explain each step of a construction.
Summary
✓ SSS: draw the longest side, then two arcs.
✓ SAS: draw one side, measure the angle, then measure the second side.
✓ The perpendicular bisector cuts a segment in half at 90°.
✓ Always leave your construction lines.
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EXAM FOCUS Using ruler and compasses only, construct the perpendicular bisector of the line AB. You must show all your construction lines. (2 marks) Use the same compass setting from both ends, more than half of AB. Do not rub out the arcs. |