Answer to Question #143390 in Trigonometry for mike

Question

Two fire spotting towers are 17 km apart on an east-west line. From Tower A, a fire is seen
on a bearing of 130°. From Tower B, the same fire is spotted on a bearing of 200°. Which
tower is closest to the fire and how far is that tower from the fire? Draw a labelled diagram.

Answer 1

$\angle BAF=130\degree-90\degree=40\degree$

$\angle ABF=270\degree-200\degree=70\degree$

$\triangle ABF: \angle AFB=180\degree-70\degree-40\degree=70\degree=\angle ABF,$ so

$AF=AB=17$ km.

$\frac{BF}{sin\angle BAF}=\frac{AB}{sin\angle AFB},$ $BF=17\cdot\frac{sin40\degree}{sin70\degree}\approx11.6$ km.

Tower B is closest to the fire and its distance is 11.6 km.