IGCSE Maths / Geometric constructions
Back bearings and return journeys revision notes
Edexcel International GCSE Maths (4MA1), chapter 13: Geometric constructions, specification point 4.4d. Free lesson.
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A back bearing gives the direction along the same straight line in the opposite direction. You can find it by adding or subtracting 180°, then use it to work out angles on a journey with more than one leg.
Why back bearings are 180° apart
A bearing is measured clockwise from north. A back bearing describes the same line in the opposite direction, measured from north at the far end. The north lines at the two places point the same way, so they are parallel. The line between the places crosses both north lines, creating co-interior angles, which add to 180°. For a bearing of B from A of 070°, the co-interior angle at B is 180° − 70° = 110°. That 110° is measured anticlockwise from north at B. The clockwise bearing is the rest of the turn: 360° − 110° = 250°. So the bearing of A from B is 250°, which is 180° away from…
How to choose add or subtract
A bearing must be between 000° and 360°. Of the two possible calculations, use the one that keeps the answer in that range. This gives a quick rule for finding a back bearing. If the given bearing is under 180°, add 180°. If the given bearing is 180° or more, subtract 180°. Keep three figures in the answer. For example, write 35° as 035°.
Check that the answer is a valid bearing
A result below 000° or above 360° cannot be a bearing. If that happens, you chose the wrong operation, so switch from adding to subtracting or the other way round. For 215°, adding 180° gives 395°, which is past a full turn. Instead, subtract 180°: 215° − 180° = 035°.
Use a back bearing to find a turning angle
For an angle at a turning point, first find the direction back along the first leg. Then both directions are bearings measured from the same north line, so the angle between them is their difference.
Apply the method to a return journey
If the bearing of X from Y is 246°, it is at least 180°, so subtract 180° to find the bearing of Y from X. The answer is 066°.