Question 1112902: A ranger in tower A spots a fire at a direction of 352 degrees. A ranger in tower B, located 45 mi at a direction of 41 degrees from tower A, spots the fire at a direction of 294 degrees. How far from tower A is the fire? How far from tower B?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Direction angles (typically) are measured clockwise from North, which is typically in the direction of the positive  -axis on a coordinate plane.
Thus, if the direction of the fire from A is  , then the angle measured on the coordinate plane must be 360 - 352 + 90, that is * .
The direction of B from A is  , so measured on the coordinate plane the angle is 90 - 41 = 49.
That means that the angle from the line between A and B to the line between A and the Fire must be 98 - 49 = 49 degrees. Now we have angle A.
Considering a line parallel to the x-axis through point B, measured in the coordinate plane would be 360 - 294 + 90 = 156. Continuing clockwise, the angle from the line between B and the Fire and the horizonal line through B is 180 - 156 = 24. By opposite interior angles of a transversal of two parallel lines, the angle from the line between A and B and the horizontal line through B is 49 degrees. Hence, the angle from the line between A and B and the line between B and the Fire is 24 + 49 = 73 degrees.
Then the final angle of the triangle, at the point of the fire is 180 - (73 + 49) = 66 degrees.
Use the Law of Sines:
}\ =\ \frac{\overline{AF}}{\sin(73)}\ =\ \frac{\overline{BF}}{\sin(49)})
The rest is just calculator work.
John
My calculator said it, I believe it, that settles it
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