Select the correct answer. Fire station B is 15 miles due east of fire station A. Firefighters at station A spot a fire at N 60 ∘ E, or 30 ∘ . Firefighters at station B spot the same fire at N 40 ∘ W, or 320 ∘ . A right triangle of corners fire, Station B and station A were the length between station A and station B is 15 miles and the angle at station A is 30 degrees and at B is 50 degrees What is the approximate distance between station B and the fire? 7.6 miles 6.4 miles 11.7 miles 29.5 miles

You want the distance from Station B to a fire spotted at a bearing 320°, if it is 15 mi east of Station A, where the bearing to the fire is 60°.

Triangle

The triangle with vertices A, B, F has internal angle A = 90° -60° = 30°, internal angle B = 320° -270° = 50°, and internal angle F = 180° -(30° +50°) = 100°.

Side AB is given as 15 miles.

Law of Sines

The law of sines tells you the relationship between sides and angles is ...

a/sin(A) = f/sin(F)

a = f·sin(A)/sin(F) = (15 mi)·sin(30°)/sin(100°) ≈ 7.6 mi

The distance from Station B to the fire is about 7.6 miles.

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Additional comment

Bearing angles are measured CW from north. Station B is at a bearing of 90° from Station A, and Station A is at a bearing of 270° from Station B. The triangle's internal angles are the (positive) difference between the direction to the fire and the direction to the other station.

We consider the triangle joining stations A, B and the Fire to be triangle ABF. Side f is AB, opposite vertex F.