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Maths · Transformations and constructions

Constructions 1

Construct triangles from their sides or from angles and sides, and construct the perpendicular bisector of a line, using compasses and a straight edge.

  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What do the angles in a triangle add up to?

    Show answerHide answer

    \(180^\circ\)

  2. 2

    What is a perpendicular line?

    Show answerHide answer

    One that meets another at \(90^\circ\)

  3. 3

    What does bisect mean?

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    Cut exactly in half

  4. 4

    What is the longest side of a right-angled triangle called?

    Show answerHide answer

    The hypotenuse

  5. 5

    What is the midpoint of a line 8 cm long?

    Show answerHide answer

    The point 4 cm from each end

Learning Objectives

  1. 1Construct a triangle given three sides (SSS).
  2. 2Construct a triangle given two sides and the angle between them (SAS).
  3. 3Construct the perpendicular bisector of a line segment.
  4. 4Explain 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 an SSS Triangle

For a triangle with sides 6 cm, 5 cm and 4 cm.

  1. 1 Draw the longest side

    Measure 6 cm with a ruler and label the ends A and B.
  2. 2 Set the compasses to 5 cm

    Put the point at A and draw an arc above the line.
  3. 3 Set the compasses to 4 cm

    Put the point at B and draw a second arc crossing the first.
  4. 4 Mark C where the arcs cross

    Join C to A and C to B with a ruler.
  5. 5 Label and check

    Measure to check each side and keep the arcs.

Can the Triangle Be Made?

Can a triangle be constructed with sides 3 cm, 4 cm and 8 cm?

Show the solutionHide the solution
  1. 1 The two shorter sides must add up to more than the longest \(3 + 4 = 7\)
  2. 2 Compare with the longest side \(7 < 8\)
  3. 3 The arcs would not meet The two short sides cannot reach each other

AnswerNo: \(3 + 4 < 8\), so the arcs do not cross and no triangle exists.

An SAS Triangle

Construct a triangle with sides of 5 cm and 7 cm and an angle of \(40^\circ\) between them.

Show the solutionHide the solution
  1. 1 Draw the 7 cm side Label the ends A and B
  2. 2 Measure \(40^\circ\) at A Use a protractor and mark the angle
  3. 3 Draw a line from A through the mark Measure 5 cm along it and label C
  4. 4 Join B to C The triangle ABC is complete

AnswerTriangle ABC with AB = 7 cm, AC = 5 cm and angle A = \(40^\circ\).

Constructing the Perpendicular Bisector

The same radius from both ends.

  1. 1 Open the compasses more than half of AB

    Any radius bigger than half will do
  2. 2 Draw arcs from A above and below the line

    Keep the same radius
  3. 3 Draw arcs from B, crossing the first arcs

    Do not change the radius
  4. 4 Mark the two crossing points

    One above and one below AB
  5. 5 Join the crossing points with a ruler

    This line bisects AB at \(90^\circ\)

Which Construction?

SSS triangle

  • 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.

SAS triangle

  • You know two sides and the angle between them.
  • Use a protractor for the angle.
  • The third side is found by joining the ends.

Construct and Measure

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...?

  1. 1Draw a line of exact length.
  2. 2Construct an SSS triangle.
  3. 3Construct an SAS triangle.
  4. 4Use a protractor accurately.
  5. 5Construct a perpendicular bisector.
  6. 6Keep construction arcs.
  7. 7Test whether three lengths make a triangle.
  8. 8Explain each step of a construction.

Summary & Exam Focus

  • 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 \(90^\circ\).
  • Always leave your construction lines.

Exam focus

Using ruler and compasses only, construct the perpendicular bisector of the line AB. You must show all your construction lines. (2 marks) (2 marks)

Use the same compass setting from both ends, more than half of AB. Do not rub out the arcs.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Construct
Draw accurately using only a ruler and compasses (a protractor when an angle is given).
Compasses
An instrument for drawing arcs and circles.
Arc
Part of the circumference of a circle.
Bisect
To cut exactly in half.
Perpendicular bisector
A line at \(90^\circ\) through the midpoint of a segment.
Vertex
A corner of a shape.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Construct 3 marks

    Using ruler and compasses only, construct a triangle with sides of length 6 cm, 5 cm and 4 cm. Show all your construction lines.

    Show answerHide answer

    Model answer

    Draw a line of 6 cm. From one end draw an arc of radius 5 cm and from the other draw an arc of radius 4 cm, crossing above the line. Join the crossing to each end.

    Mark scheme

    • A 6 cm line drawn accurately — B1
    • Both arcs drawn correctly — M1
    • Triangle completed and arcs shown — A1
  2. Question 2 Construct 2 marks

    The line AB is 6 cm long. Using ruler and compasses only, construct the perpendicular bisector of the line AB. Show all your construction lines.

    A horizontal line AB with plenty of space around it.
    Show answerHide answer

    Model answer

    Equal arcs from A and B, with radius more than 3 cm, crossing above and below; a straight line through the two crossings.

    Mark scheme

    • Arcs of equal radius from A and B — M1
    • Correct crossings, both sides — A1
  3. Question 3 Explain 2 marks

    Explain why it is not possible to construct a triangle with sides of length 3 cm, 4 cm and 8 cm.

    Show answerHide answer

    Model answer

    The two shorter sides add up to \(3 + 4 = 7\) cm, which is less than 8 cm, so the arcs would not meet.

    Mark scheme

    • \(3 + 4 = 7\) — M1
    • \(7 < 8\), so the arcs do not meet — C1
  4. Question 4 Construct 3 marks

    Construct a triangle ABC with AB = 7 cm, AC = 5 cm and angle \(BAC = 40^\circ\).

    Show answerHide answer

    Model answer

    Draw AB = 7 cm. Measure \(40^\circ\) at A. Mark C 5 cm from A along the line. Join B to C.

    Mark scheme

    • AB drawn accurately — B1
    • Angle of \(40^\circ\) at A — M1
    • AC = 5 cm and triangle completed — A1
  5. Question 5 Measure 2 marks

    In the triangle with sides 6 cm, 5 cm and 4 cm, measure the size of the largest angle.

    Show answerHide answer

    Model answer

    The largest angle is opposite the 6 cm side and is about \(83^\circ\) (accept \(81^\circ\) to \(85^\circ\)).

    Mark scheme

    • Identifying the angle opposite the 6 cm side — M1
    • 83 (within the range) — A1
  6. Question 6 Decide 2 marks

    For each set of lengths, state whether a triangle can be constructed. (a) 5 cm, 6 cm, 7 cm (b) 2 cm, 3 cm, 6 cm

    Show answerHide answer

    Model answer

    (a) Yes, because \(5 + 6 = 11 > 7\). (b) No, because \(2 + 3 = 5 < 6\).

    Mark scheme

    • (a) Yes — B1
    • (b) No — B1

Quick check

  1. Which instrument draws an arc?

    1. AA ruler
    2. BA protractor
    3. CCompasses
    4. DA set square
    Show answerHide answer

    C: Compasses

    Compasses draw arcs and circles.

  2. The perpendicular bisector of AB meets AB at...

    1. AA
    2. BThe midpoint, at 90°
    3. CB
    4. DAny point
    Show answerHide answer

    B: The midpoint, at 90°

    The midpoint, at a right angle.

  3. Should you rub out construction arcs?

    1. ANo
    2. BYes
    3. COnly some
    4. DOnly the first
    Show answerHide answer

    A: No

    No: they show your method and earn marks.

  4. Can a triangle have sides 4 cm, 4 cm and 9 cm?

    1. AYes
    2. BOnly if isosceles
    3. COnly with a protractor
    4. DNo
    Show answerHide answer

    D: No

    \(4 + 4 = 8 < 9\), so the arcs do not meet.

  5. To construct an SAS triangle you need a...

    1. ASet square
    2. BProtractor
    3. CCalculator
    4. DPair of scissors
    Show answerHide answer

    B: Protractor

    A protractor for the angle between the two sides.

  6. Every point on the perpendicular bisector of AB is...

    1. ACloser to A
    2. BCloser to B
    3. CEquidistant from A and B
    4. DOn the line AB
    Show answerHide answer

    C: Equidistant from A and B

    The same distance from A and from B.

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