Mathematics (Core)Exam code: 0580

Bearings, Constructions and Scale Drawings

Geometry

PractiseFlashcards

The three rules

  • A bearing describes a direction, and it follows three rules without exception
    • Measured from north
    • Measured clockwise
    • Written with three figures, so 59° is recorded as 059°
  • Bearings therefore run from 000° round to 360°
  • The compass directions have fixed bearings: north 000°, east 090°, south 180°, west 270°, with north-east 045°, south-east 135°, south-west 225° and north-west 315°
A compass rose with each of the eight directions labelled with its three-figure bearing: north 000 or 360, north-east 045, east 090, south-east 135, south 180, south-west 225, west 270 and north-west 315, read clockwise from north
Source: Bearings by Save My Exams
  • "Due east" means exactly 090°, and the word "due" signals an exact compass direction

Which point to start from

  • "The bearing of B from A" means stand at A and look towards B, so the north line is drawn at A
  • The point after "from" is where the north line goes, and getting this the wrong way round reverses the answer
  • To measure one, start by drawing north at wherever you are standing, join the two points, and take the clockwise angle with a protractor
  • To plot one, draw the north line, measure the angle clockwise, rule a long line, then mark the distance along it using the scale
A scale drawing at 1 cm to 10 km with its own north arrow: a red north line has been drawn at the point P and a long red line ruled away from P at the measured clockwise angle from that north line
Source: Bearings by Save My Exams

Worked example

Measuring and plotting a bearing

The scale drawing shows two ports, A and B, with a north line drawn at A. The scale is 1 cm to 30 km. Measure the bearing of B from A. A lighthouse, L, is 120 km from A on a bearing of 235°. Mark the position of L on the drawing.

A scale drawing with a north line drawn straight up from point A and a ruled line from A to point B, which lies up and to the right of A. The angle at A is not marked. A note reads: scale 1 cm to 30 km.

Solution:

  • The bearing of B from A is measured at A, so use the north line drawn there
  • Put the centre of the protractor on A with 0° on the north line, and read the angle clockwise round to the line AB: 64°
  • Bearings are written with three figures, so the bearing of B from A is 064°
  • For L, change the real distance into a length on the drawing first: 120 ÷ 30 = 4 cm
  • 235° is more than 180°, so extend the north line down through A to point south, and measure 235 − 180 = 55° clockwise from south (or 360 − 235 = 125° anticlockwise from north)
  • Rule a line from A at that angle and mark the point 4 cm from A along it: that point is L
The same scale drawing with L plotted: a dashed line runs south from A, and a ruled line 4 cm long goes from A down and to the left to L, with the 55° between the south line and AL marked; the 64° between the north line and AB is marked too
  • Check: a bearing between 180° and 270° points south-west, and L lies below and to the left of A

Back bearings

  • The bearing of A from B and the bearing of B from A always differ by exactly 180°
  • Reverse a bearing below 180° by adding on a further 180°
  • Reverse one above 180° by taking 180° away instead, which keeps the answer inside the 000° to 360° range
  • The rule follows from the two north lines being parallel: the angle at A between north and AB and the angle at B between north and BA are co-interior, so they add up to 180°, and that puts the two bearings exactly 180° apart
A two-leg journey drawn to scale: a ship sails from P along one bearing to Q, then a fresh north line is drawn at Q and the second leg is ruled from there to the final position, each north line drawn at the point being set off from
Source: Bearings by Save My Exams

Worked example

Reversing a bearing both ways

The bearing of B from A is 072°. The bearing of Q from P is 295°. Find the bearing of A from B, and the bearing of P from Q.

Points A and B joined by a straight line, with a dashed north arrow drawn at each of them so the two north lines are parallel. The angle at A, measured clockwise from its north line round to the line AB, is marked 072 degrees. The angle at B is not marked. The figure is marked NOT TO SCALE.

Solution:

  • For the first, 072° is less than 180°, so add 180°
  • 72 + 180 = 252, so the bearing of A from B is 252°
  • For the second, 295° is more than 180°, so subtract 180°
  • 295 − 180 = 115, so the bearing of P from Q is 115°
  • Both answers are written with three figures and lie between 000° and 360°
  • A question may give one bearing as an expression and the other as a number, which turns the same rule into an equation to solve
  • Harder bearings questions combine this with Pythagoras or trigonometry, so draw and label a clear diagram before calculating

Worked example

A bearing from angles on a diagram

Points A, B and C are shown in the diagram. B is on a bearing of 118° from A, and angle ABC = 75°. Find the bearing of C from B.

Points A, B and C joined by the lines AB and BC, with a dashed north arrow at A and at B. At A the angle measured clockwise from north round to the line AB is marked 118 degrees; at B the angle between the lines BA and BC is marked 75 degrees. C lies below and to the left of B. The figure is marked NOT TO SCALE.

Solution:

  • The bearing is measured from B, so work with the north line at B
  • The north lines at A and B are parallel, so the bearing of A from B is the reverse of 118°: 118 + 180 = 298°
  • Seen from B, the line to C is a further 75° anticlockwise from the line to A, so subtract: 298 − 75 = 223
  • The bearing of C from B is 223°
  • Check another way: the angle between north at B and BA is co-interior with the 118° at A, so it is 180 − 118 = 62°, and the clockwise angle from north round to BC is 360 − 62 − 75 = 223°
  • C lies to the south and west of B, so its bearing must be between 180° and 270°, and 223° lies in that range

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