WAEC Mathematics 2010 Theory question 109

Theory question 109

Bearings

In the diagram, |AB| = 8km, |BC| = 13km, the bearing of A from B is 310° and the bearing of B from C is 230°. Calculate, correct to 3 significant figures, (a) the distance AC; (b) the bearing of C from A; (c) how far east of B, C is.

Show the model answer

Model answer

Bearing of C from B = 230° − 180° = 050°, so ∠ABC = 360° − 310° + 50° = 100°.
(a) AC² = 8² + 13² − 2(8)(13) cos 100° = 233 + 36.1 = 269.1, so AC ≈ 16.4 km.
(b) sin ∠BAC = 13 sin 100°/16.40 ≈ 0.780, so ∠BAC ≈ 51.3°. Bearing of B from A = 130°, so bearing of C from A = 130° − 51.3° ≈ 078.7°.
(c) 13 sin 50° ≈ 9.96 km east of B.

Explanation

Find the angle at B from the two bearings, then use the cosine rule.