Posted by Mike on Tuesday, May 10, 2011 at 2:25pm.
Two fire towers are 30 kilometers apart, tower A being due west of tower B. A fire is spotted from the towers, and the bearings from A and B are E 14 degrees N and W 34 degrees N, respectively. Find the distance d of the fire from the line segment AB.

Math  Henry, Wednesday, May 11, 2011 at 4:37pm
1. Draw line segment AB.
2. Draw the 14degree angle from point
A.
3. Draw the 34degree angle from point B. Label the intersection of these 2 lines point C. Now we have formed
triangle ABC.
4. Draw the altitude from point C perpendicular to AB and label it CD.
The distance of the fire from AB is
equal to the altitude(CD).
A + B + C = 180 Deg.
14 + 34 + C = 180,
C = 132 Deg.
a/sinA = c/sinC,
a/sin14 = 30/sin132,
Multiply both sides by sin14:
a = 30sin14 / sin132 = 9.77km.
CD = 9.77sin34 = 5.46km = dist. from
fire to AB.

Math  sam, Wednesday, December 4, 2013 at 5:50am
21.9

Math  Arpita, Thursday, November 5, 2015 at 2:02am
1. Call the point where d intersects AB point C.
2. Let CB equal x.
3. cot(14)= (30x)/d
cot(34)= x/d
4. cot(14)= (30/d) (x/d)
cot(14)= (30/d) cot(34)
cot(14)+ cot(34)= (30/d)
d(cot14+ cot34)= 30
d = 30/ (cot14+ cot34)
d = 5.46 km

Math  Arpita, Thursday, November 5, 2015 at 2:05am
Sorry this one's easier to read.
1..
Call the point where d intersects AB point C.
2..
Let CB equal x.
3..
cot(14)= (30x)/d
cot(34)= x/d
4..
cot(14)= (30/d) (x/d)
cot(14)= (30/d) cot(34)
cot(14)+ cot(34)= (30/d)
d(cot14+ cot34)= 30
d = 30/ (cot14+ cot34)
d = 5.46 km
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