Triangle word problems? I just need help with this first one to show me how to set it up?

Question

Here is the question...
Two fire look out stations are 20 miles apart with station B directly east of station A. Both stations spot a fire. The bearing of the fire from station A is N60 degrees east and the bearing of the fire from station B is N40 degrees west. How far is it from Station A to the Fire?


In miles..
This is a multiple choice
a) 7.5 miles
b) 1.5 miles
c) 16 miles
d) 10.2 miles
Thanks for any help you can give..

Answer 1

If you draw a line straight north from station A, this line makes a 90 degree angle with the line from station A to station B, which is due east of station A. The line from station A to the fire makes a 60 degree angle with the line straight north from station A on its east side (this is what N60 degrees east means). Therefore the angle between the lines from station A to the fire and from station A to station B is 30 degrees.
Now draw a line straight north from station B. This line also makes a 90 degree angle with the line from station A to station B. The line from station B to the fire makes a 40 degree angle with the line straight north from station B, on its west side (N40 degrees west). Therefore the angle between the lines from station B to the fire and from station B to station A is 50 degrees.
Now look at the triangle formed by station A, station B, and the fire. The angle at A is 30 degrees and the angle at B is 50 degrees, so the angle at the fire (between the lines to station A and to station B) is 100 degrees, since the sum of the angles of a triangle is 180 degrees. The distance from station A to station B is 20 miles. Let d stand for the distance from station A to the fire. The line from station A to station B is opposite the 100 degree angle. The line from station A to the fire is opposite the 50 degree angle. So by the law of sines
d / sin 50 = 20 / sin 100, or d = (sin 50 / sin 100)*20 miles
By my calculator, sin 50 = .76604, sin 100 = .98481,
sin 50 / sin 100 = .77786, and d = (.77786)(20 miles) = 15.56 miles. This rounds to 16 miles, which is your choice (c).
d / sin 50 = 20 / sin 100

Answer 2

Draw a picture and use the law of sines to get the answer. The angles for the triangle formed are 50, 30, and 100 degrees. Side opposite of the 100 degree is the 20 miles and the side opposite the 50 degree is the unknown. Use the law of sines to get the answer.

b/sin50 = 20/sin100
when I worked this out I came up with c for the answer. 15.55 miles which rounds off to 16 miles. You might work it out and make sure I did all the correct calculations.

Answer 3

This is a sine rule question.

I hope you've drawn the diagram, because I can't send you one.

The angle opposite AB is 100 deg (angle sum of triangle), and the length you wish to find is opposite to the 50 deg (complement of 40 deg) at B.

So x/(sin 50 deg) = 20/(sin 100 deg)
and x = 20*(sin 50 deg)/sin(100 deg).

I don't know which of the answers this gives, but I'm sure you can do it on your calculator.