Teach Me Maths

Constructions

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How to construct a perpendicular bisector, an angle bisector, perpendiculars at and from a point, and triangles from three sides with a ruler and compasses, leaving every arc showing as the exam expects.

Constructions in the exam

A construction is an accurate drawing made with only a ruler and a pair of compasses. The ruler draws straight lines and the compasses draw arcs. You don't use a protractor, and you don't measure to find a halfway point.

In the exam a question says "Use ruler and compasses to construct …", usually for 2 marks, and adds "You must show all your construction lines". The arcs are your method. A line in the right place with no arcs gets at most 1 of the 2 marks.

The constructions you need are:

  • the perpendicular bisector of a line
  • the bisector of an angle
  • a perpendicular at a point on a line
  • a perpendicular from a point to a line
  • a triangle from its three sides, including an equilateral triangle with its 60^\circ angles

You can't draw on this page, so the "your turn" steps ask about the decisions: what radius to use, where the arcs cross, and the lengths and angles that come out. Practise each construction on paper with a sharp pencil as you read.

Arcs and the compass opening

The distance from the point of the compasses to the pencil is the radius. Every point on an arc is the same distance from its centre, so two points on one arc are the same distance from the centre.

Most constructions draw two arcs from two different points with the same opening. Where the arcs cross is the same distance from both points. Don't change the opening between the two arcs.

For arcs from A and B to cross, the opening must be more than half of AB. When AB is 8 cm, half of AB is 4 cm. An opening of 3 cm leaves a gap, but 5 cm works.

The perpendicular bisector of a line

The perpendicular bisector of AB cuts AB into two equal halves and crosses it at 90^\circ. Every point on it is the same distance from A as from B.

  • Open the compasses to more than half of AB.
  • With the point on A, draw an arc above AB and an arc below it.
  • Keep the same opening. With the point on B, draw two arcs that cross the first two.
  • Join the two crossing points with a ruler.

Worked example 1: Bisecting a line

AB is a line 9 cm long. Use ruler and compasses to construct the perpendicular bisector of AB. You must show all your construction lines.

  1. Your turn. The opening must be more than half of AB. Work out half of 9 cm.

    Half of AB is 9\div2=4.5 cm

  2. Open the compasses to more than 4.5 cm, such as 6 cm. With the point on A, draw an arc above AB and an arc below it.
  3. Keep the opening at 6 cm. With the point on B, draw two arcs that cross the first two at P and Q.
  4. Join P and Q with a ruler, through AB. Leave all four arcs showing.
  5. Answer: PQ is the perpendicular bisector. It meets AB at 90^\circ at its midpoint M, with AM=MB=4.5 cm

The bisector of an angle

The bisector of an angle cuts it into two equal angles. Bisecting an angle of 80^\circ makes two angles of 40^\circ.

  • With the point on the corner of the angle, draw an arc that crosses both lines of the angle.
  • Put the point on each of those two crossing points in turn. With the same opening, draw two arcs that cross inside the angle.
  • Join the corner to where those arcs cross, with a ruler.

The first arc must be centred on the corner, so the two crossing points are the same distance from it. Arcs started anywhere else give a line that is not the bisector.

Worked example 2: Bisecting an angle

Angle PQR is 112^\circ. Use ruler and compasses to construct the bisector of angle PQR. What size is each of the two angles it makes?

  1. Put the compass point on Q, the corner. Draw an arc that crosses QR at C and QP at D.
  2. Put the point on C, then on D, with the same opening. Draw two arcs that cross inside the angle at X.
  3. Join Q to X with a ruler, and carry the line on. Leave the arcs showing.

  4. Your turn. The bisector splits 112^\circ into two equal angles. What size is each one?

    Each angle is 112\div2=56^\circ

  5. Answer: angle PQX= angle XQR=56^\circ

Perpendiculars at and from a point

Perpendicular means at 90^\circ. Both of these constructions start with an arc centred at P that crosses the line twice, at C and D. Then you finish as if bisecting CD.

At a point P on the line. Draw an arc centred at P that crosses the line at C and D. Open the compasses wider. Draw arcs from C and D that cross at E, above the line. Join P to E.

From a point P above the line. Draw an arc centred at P that crosses the line at C and D. With the same opening, draw arcs from C and D that cross at Q, below the line. Join P to Q. It meets the line at R.

PR is the shortest distance from P to the line. The line you construct must go through P, so start with the compass point on P. Bisecting the line between its labelled ends gives a line that misses P.

Worked example 3: A right angle at a point

On a plan, a shelf bracket must meet the wall line AB at right angles, at the point P on AB. Use ruler and compasses to construct the line through P that is perpendicular to AB.

  1. Put the compass point on P. Draw an arc of radius 3 cm that crosses the line on both sides of P, at C and D.

  2. Your turn. PC and PD are both 3 cm. How long is CD, in centimetres?

    CD=3+3=6 cm

  3. For arcs from C and D to cross, open the compasses to more than half of CD, which is more than 3 cm. Use 5 cm.
  4. With the point on C, then on D, draw arcs above the line. They cross at E.
  5. Join P to E with a ruler. Leave all the arcs showing.
  6. Answer: the line PE, with angle EPB=90^\circ

Worked example 4: A perpendicular on a grid

P is the point (2,4). The perpendicular from P to the x-axis is constructed with ruler and compasses. The first arc, centred at P, crosses the x-axis at C(-1,0) and D(5,0). The perpendicular meets the x-axis at R. Work out the coordinates of R.

  1. C and D are on one arc centred at P, so PC=PD. Here both are 5 units.
  2. With the same opening, 5 units, arcs from C and D cross below the axis at Q(2,-4). Join P to Q.
  3. PQ is the perpendicular bisector of CD, so R is halfway between C and D.

  4. Your turn. Give the coordinates of R.

    Write your answer as (x, y)

    R is halfway between x=-1 and x=5, on the x-axis: R=(2,0)

  5. Answer: R=(2,0). The shortest distance from P to the x-axis is PR=4 units.

Triangles from three sides

To construct a triangle from its three sides, draw the longest side with a ruler. Draw an arc from each end, using the other two sides as the radii. The arcs cross at the third corner. Join it to both ends.

The two shorter sides must add up to more than the longest side, or the arcs don't meet. Sides of 3 cm, 4 cm and 9 cm make no triangle, because 3+4=7, which is less than 9.

An equilateral triangle has three equal sides, so both arcs use the length of AB as the radius. Each angle is 60^\circ, so this is also how you construct an angle of 60^\circ.

Bisecting gives more angles. Bisect a constructed 60^\circ angle to get 30^\circ, and bisect a constructed 90^\circ angle to get 45^\circ.

Worked example 5: A triangle from three sides

Use ruler and compasses to construct triangle ABC, where AB=8 cm, AC=6 cm and BC=5 cm.

  1. Check first: 5+6=11, which is more than 8, so the arcs will meet.
  2. Draw AB, 8 cm long, with a ruler. Label the ends A and B.

  3. Your turn. Corner C is the end of side AC. What radius, in centimetres, do you use for the arc centred at A?

    C is 6 cm from A, so the arc centred at A has radius 6 cm

  4. C is 5 cm from B, so draw an arc of radius 5 cm centred at B. The two arcs cross at C.
  5. Join C to A and to B with a ruler. Leave both arcs showing.
  6. Answer: triangle ABC, with both arcs at C left showing.

Worked example 6: The same distance from two places

On a map grid, 1 square represents 1 km. Village B is at (1,2) and village C is at (9,2). A phone mast T must be the same distance from B as from C, and exactly 5 km from B. T is north of the line BC. Work out the coordinates of T.

  1. T is the same distance from B as from C, so T is on the perpendicular bisector of BC.
  2. BC is horizontal, with midpoint M(5,2), so its perpendicular bisector is the vertical line x=5
  3. T is 5 km from B, so T is on an arc of radius 5 squares centred at B. The arc crosses the line x=5 at two points.

  4. Your turn. T is the crossing point north of BC. Give the coordinates of T.

    Write your answer as (x, y)

    T=(5,5), which is 3 squares above M

  5. Check with Pythagoras: BM=4 and MT=3, and 4^2+3^2=16+9=25=5^2, so BT=5 km.
  6. Answer: T=(5,5)

Common mistakes

  • Leaving out the construction arcs. A line that looks right but has no arcs gets at most 1 of the 2 marks, so draw every arc and never rub them out.
  • Measuring instead of constructing. Don't measure AB with a ruler to find the middle, or use a protractor for a 90^\circ angle; use the compasses.
  • Changing the compass opening. The two arcs from A and B, or from C and D, must use the same opening.
  • Opening the compasses too little. Arcs from A and B only cross when the opening is more than half of AB.
  • Bisecting the wrong line. For a perpendicular from P, put the compass point on P first, so the line goes through P.
  • Coming to the exam without compasses. Freehand arcs are hard to get accurate, so bring a pair of compasses that holds its opening.

Practice questions

More on this topic: Constructions worksheet with full solutions