WAEC Mathematics 2002 Theory question 111

Theory question 111

Bearings

In the diagram, |PQ| = 8m, |QR| = 13m, the bearing of Q from P is 050° and the bearing of R from Q is 130°. (a) Calculate, correct to 3 significant figures, (i) |PR| (ii) the bearing of R from P. (b) Calculate the shortest distance between point Q and PR, hence the area of triangle PQR.

Show the model answer

Model answer

(a)(i) ∠PQR = 50° + (180° − 130°) = 100°.
PR² = 8² + 13² − 2(8)(13) cos 100° = 233 + 36.1 = 269.1, so PR ≈ 16.4 m.
(ii) sin ∠QPR = 13 sin 100°/16.40 = 0.7805, so ∠QPR ≈ 51.3°. Bearing of R from P = 50 + 51.3 ≈ 101°.
(b) Shortest distance = perpendicular from Q to PR = 8 sin 51.3° ≈ 6.24 m.
Area = ½ × 16.4 × 6.24 ≈ 51.2 m² (check: ½ × 8 × 13 × sin 100° = 51.2).

Explanation

Use the north lines at P and Q to find the angle at Q first.