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Maths · Transformations and constructions

Constructions 2

Bisect an angle, drop a perpendicular from a point to a line, and construct angles of 90°, 60° and 30° using compasses.

  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What does perpendicular mean?

    Show answerHide answer

    At \(90^\circ\)

  2. 2

    What is half of \(60^\circ\)?

    Show answerHide answer

    \(30^\circ\)

  3. 3

    How many degrees in each angle of an equilateral triangle?

    Show answerHide answer

    \(60^\circ\)

  4. 4

    What is an angle bisector?

    Show answerHide answer

    A line that cuts an angle in half

  5. 5

    What are the tools for a construction?

    Show answerHide answer

    Ruler, compasses (and a sharp pencil)

Learning Objectives

  1. 1Construct the bisector of an angle.
  2. 2Construct a perpendicular from a point to a line.
  3. 3Construct a perpendicular at a point on a line.
  4. 4Construct angles of \(60^\circ\), \(30^\circ\) and \(90^\circ\).

Bisecting an Angle

Three arcs and one line.

  1. 1 Arc from the vertex

    Centre B, cutting both arms at P and Q
  2. 2 Arcs from P and Q

    Same radius from each; they cross at R
  3. 3 Draw the line BR

    This is the angle bisector
  4. 4 Check

    Measure the two halves; they should be equal

Angle Bisector Check

Angle ABC is \(70^\circ\). After bisecting it, what is the size of each half?

Show the solutionHide the solution
  1. 1 The bisector cuts the angle in half \(70 \div 2\)
  2. 2 Work out \(35\)

AnswerEach half is \(35^\circ\).

Perpendicular from a Point to a Line

Point P is above the line.

  1. 1 Arc centred on P

    It crosses the line at two points, X and Y
  2. 2 Arcs from X and Y

    Same radius, crossing on the other side of the line at Z
  3. 3 Join P to Z

    This line is perpendicular to the line
  4. 4 Label the right angle

    Mark it with a small square

Perpendicular at a Point on a Line

Point Q is on the line.

  1. 1 Arc centred on Q

    It crosses the line either side at U and V
  2. 2 Larger arcs from U and V

    Same radius, crossing above the line at W
  3. 3 Join Q to W

    This line is perpendicular to the line at Q
  4. 4 Check

    The angle is \(90^\circ\)

Constructing 60° and 30°

Compasses can make an equilateral triangle, which gives \(60^\circ\).

  • 60°

    Draw an arc from A across the line at B, then an arc of the same radius from B; join A to where they cross.

  • 30°

    Bisect the \(60^\circ\) angle.

  • 90°

    Use the perpendicular construction.

  • 45°

    Bisect a \(90^\circ\) angle.

Build a Kite

Construct an angle of \(60^\circ\) at A. Bisect it to make \(30^\circ\). Then use the perpendicular construction to make a right angle at a point on one arm. Sketch a kite using your construction lines.

1. Make 60° with compasses.

2. Bisect it.

3. Add a 90° angle.

A good answer shows: A \(60^\circ\) angle from an equilateral construction, \(30^\circ\) from the bisector, and a \(90^\circ\) angle by the perpendicular at a point. A kite can be made from two triangles sharing the bisector.

Can I...?

  1. 1Bisect an angle.
  2. 2Drop a perpendicular from a point.
  3. 3Draw a perpendicular at a point on a line.
  4. 4Construct \(60^\circ\).
  5. 5Construct \(30^\circ\).
  6. 6Construct \(90^\circ\).
  7. 7Explain why the constructions work.
  8. 8Show all construction lines.

Summary & Exam Focus

  • Angle bisector: arc, two equal arcs, join to the vertex.
  • Perpendicular from a point: arc, two equal arcs, join.
  • \(60^\circ\) from an equilateral triangle; halve it for \(30^\circ\).
  • Keep all construction arcs.

Exam focus

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

Start with an arc centred on the vertex, then use equal arcs from the two points where it crosses the arms.

Key terms

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

Angle bisector
A line that cuts an angle exactly in half.
Perpendicular
At \(90^\circ\) to a line.
Arc
Part of a circle's circumference.
Equidistant
The same distance from two things.
Construction lines
The arcs and lines you leave to show your method.
Vertex
The point where the arms of an angle meet.

Practice questions

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

  1. Question 1 Construct 2 marks

    The diagram shows an angle ABC. Using ruler and compasses only, construct the bisector of angle ABC. Show all your construction lines.

    An angle ABC of about 70 degrees with blank space inside it.
    Show answerHide answer

    Model answer

    An arc from B crossing both arms, two equal arcs from those points crossing inside the angle, and a line from B through the crossing.

    Mark scheme

    • Arcs from B and from the two crossing points — M1
    • A correct bisector with arcs shown — A1
  2. Question 2 Construct 3 marks

    Point P is 4 cm above a horizontal line. Using ruler and compasses only, construct the perpendicular from P to the line. Show all your construction lines.

    Show answerHide answer

    Model answer

    An arc centred on P cutting the line twice; equal arcs from those two points crossing below the line; a line from P to the crossing.

    Mark scheme

    • Arc from P crossing the line twice — M1
    • Equal arcs from the two points — M1
    • A correct perpendicular — A1
  3. Question 3 Construct 3 marks

    Using ruler and compasses only, construct an angle of \(60^\circ\) at the end A of a line AB.

    Show answerHide answer

    Model answer

    An arc centred on A crossing AB; the same radius centred on that crossing; a line from A through the crossing of the arcs.

    Mark scheme

    • Arc centred on A — M1
    • Second arc with the same radius — M1
    • Line completed and angle correct — A1
  4. Question 4 Construct 3 marks

    Using ruler and compasses only, construct an angle of \(30^\circ\) at the end A of a line AB.

    Show answerHide answer

    Model answer

    Construct \(60^\circ\) at A, then bisect it to make \(30^\circ\).

    Mark scheme

    • \(60^\circ\) constructed — M1
    • Bisector constructed — M1
    • Angle of \(30^\circ\) — A1
  5. Question 5 Construct 3 marks

    Q is a point on a straight line. Using ruler and compasses only, construct the perpendicular to the line at Q.

    Show answerHide answer

    Model answer

    An arc centred on Q cutting the line on both sides; equal larger arcs from those points crossing above; a line from Q through the crossing.

    Mark scheme

    • Arc centred on Q — M1
    • Equal arcs from both points — M1
    • Correct perpendicular — A1
  6. Question 6 Explain 2 marks

    Explain why the line through the crossing arcs is the bisector of the angle in the angle bisector construction.

    Show answerHide answer

    Model answer

    The point where the arcs cross is the same distance from the two points on the arms, and both points are the same distance from the vertex, so the two triangles are congruent (a kite) and the angles are equal.

    Mark scheme

    • Equal distances used — M1
    • Congruent triangles or kite symmetry — C1

Quick check

  1. What does an angle bisector do?

    1. ADoubles the angle
    2. BCuts it in half
    3. CMakes it 90°
    4. DDraws a circle
    Show answerHide answer

    B: Cuts it in half

    It cuts the angle exactly in half.

  2. A bisected angle of 84° gives two angles of...

    1. A84°
    2. B21°
    3. C42°
    4. D168°
    Show answerHide answer

    C: 42°

    \(84 \div 2 = 42\).

  3. Which angle can be constructed directly with compasses?

    1. A60°
    2. B47°
    3. C12°
    4. D83°
    Show answerHide answer

    A: 60°

    \(60^\circ\), from an equilateral triangle.

  4. How do you construct 30°?

    1. AMeasure with a ruler
    2. BDraw a circle
    3. CBisect 90°
    4. DBisect 60°
    Show answerHide answer

    D: Bisect 60°

    Construct 60° and bisect it.

  5. In a perpendicular construction, the angle formed is...

    1. A45°
    2. B90°
    3. C60°
    4. D180°
    Show answerHide answer

    B: 90°

    A right angle: 90°.

  6. Bisecting a 90° angle gives...

    1. A30°
    2. B60°
    3. C45°
    4. D180°
    Show answerHide answer

    C: 45°

    \(90 \div 2 = 45\).

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