note 1: N 31 E means you start facing directly north, then turn 31 degrees toward the east.
note 2: N 59 W means you start facing directly north, then turn 59 degrees toward the west.
Check out the attached image to see the diagram. We start at point A, we then move to point B (see note 1 above) traveling 65 miles. Then we move to point C (see note 2) traveling 102 miles. The goal is to find the distance from A to C which is side b. Side b is opposite angle B.
I have added points D, E and F to help label the angles in a meaningful way. Angle DAB is the angle "N 31 E" mentioned earlier. Angle ABF is congruent to angle DAB (AD || EF; alternate interior angle theorem), so it is also 31 degrees. Angle EBC is the angle "N 59 W" mentioned earlier.
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We need to figure out angle ABC to be able to find the missing side.
Note that all the angles formed at point B can be combined to form a straight angle that is 180 degrees
(angle ABC) + (angle EBC) + (angle ABF) = 180
(angle ABC) + (59) + (31) = 180
(angle ABC) + 90 = 180
(angle ABC) + 90-90 = 180-90
angle ABC = 90
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We now know that angle ABC is 90 degrees
Focusing on triangle ABC, the notation "angle ABC" is the same as "angle B".
So we can use this to find side AC (aka side b). Use the law of cosines to solve for b.
b^2 = a^2 + c^2 - 2*a*c*cos(B)
b^2 = 102^2 + 65^2 - 2*102*65*cos(90)
b^2 = 102^2 + 65^2 - 2*102*65*0
b^2 = 102^2 + 65^2 - 0
b^2 = 102^2 + 65^2
b^2 = 10404 + 4225
b^2 = 14629
b = sqrt(14629)
b = 120.950403058444
b = 121 <----------- rounding to the nearest whole number
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The final answer is C) 121 miles which is approximate
