Trigonometry, bearings and elevation

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Formula

Explanation

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Worked solution

1. The hiker walks 31 m on a bearing of 068° from point A to point B. This means the direction is 68° measured clockwise from north.
2. From point B, the hiker observes point A at a bearing of 248°. This is the direction from B to A, measured clockwise from north.
3. The angle between the two bearings (068° and 248°) at point B gives the angle between the path AB and the line BA. The difference is 248° - 68° = 180°, but this is not the internal angle of triangle ABA.
4. The actual angle at point B in triangle ABA is found by considering the direction change. The bearing from B to A is 248°, so the direction from B to A is 248° - 180° = 68° relative to south. The angle between the incoming path (from A to B) and the outgoing path (from B to A) is the angle between the two directions.
5. The angle at B in triangle ABA is the difference between the reverse of the first bearing and the second bearing. The reverse of 068° is 248° (068° + 180°). The second bearing is also 248°, so the angle at B is 248° - 248° = 0°, which is not possible.
6. Correct approach: The angle at B is the difference between the direction from B to A (248°) and the direction from B to A along the path (opposite to AB). The direction from B to A along the path is 068° + 180° = 248°. So the angle between the two lines at B is 248° - 248° = 0°, which is invalid.
7. Re-evaluate: The bearing from A to B is 068°, so the direction from B to A along the path is 068° + 180° = 248°. The observed bearing from B to A is 248°, so the angle between the two lines at B is 0°, which implies points A, B, and the direction are colinear. But this contradicts the need to find a distance.
8. The correct interpretation: The hiker walks from A to B on bearing 068°, so the direction AB is 068° from north. From B, the hiker sees A at bearing 248°, which is the direction from B to A. The angle at B between the path AB and the line BA is the difference between the reverse of AB and the observed bearing.
9. The reverse of AB (from B to A along the path) is 068° + 180° = 248°. The observed bearing from B to A is 248°, so the angle between them is 248° - 248° = 0°, which again suggests colinearity.
10. This indicates that points A, B, and the direction are colinear, so the straight-line distance from A to B is simply the distance walked: 31 m. The bearing from B to A is 248°, which matches the reverse of 068°, confirming that A, B, and the path are straight.
11. Therefore, the straight-line distance from A to B is exactly 31 m, as the hiker walked directly from A to B along a straight path.
12. The distance from A to B is 31 m, and since the path is straight and the bearings confirm colinearity, no trigonometry is needed. The answer is 31 m.

Reference answer: 31 m

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