Question

I worked this problem out but I do not fully understand what I am doing.

Fire tower A is 30 kilometers due west of fire tower B.. A fire is spotted from the towers, and the bearings from A and B are N 76 degrees E and N 56 degrees W, respectively. Find the distance d of the fire from the line segment AB.

I got 14 kilometers.

Answer 1

If the fire is labeled F, then in ΔABF, the angles are

A = 14°
B = 34°

If the altitude from F to AB meets it at point P, then

h/AP = sin 14°
h/PB = sin 34°
so,
AP = h/sin14°
PB = h/sin34°
AP+PB=30

h/sin14° + h/sin34° = 30
h(1/sin14°+1/sin34°) = 30
h = 30/(1/sin14°+1/sin34°) = 5.07

I'd be interested in the steps you took to arrive at 14.