Posted by Billy Bob on Tuesday, May 22, 2012 at 6:41pm.
Never attempt a question like this without a sketch or diagram.
I labeled the position of the fire as F
and by some simple adding/subtracting of angles, I had angle A = 45° and angle B = 95°, thus angle F = 40° , and AB = 20.3
By sine law:
AF/sin95 = 20.3/sin40
AF = 20.3sin95/sin40 = appr31.46 km
BF/sin45 = 20.3/sin40
.....
the second one is quite easy,
make a sketch of the triangle, place x as the side opposite the 20° angle and (x+10) opposite the 120° angle.
by sine law :
x/sin20 = (x+10)/sin120
xsin120 = xsin20 + 10sin20
xsin120 - xsin20 = 10sin20
x(sin120 - sin20) = 10sin20
x = 10sin20/(sin120-sin20) = appr 6.527
so the smallest side is 6.527,
the largest side is 16.527
Use the sine law once more to find the third side, then add up the 3 sides.
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