Points a and b are on opposite sides of a lake. a point c is 91.791 meters from a. the measure of ∠bac ∠ b a c is 76.833°, and the measure of ∠acb ∠ a c b is determined to be 31.683°. find the distance between points a and b.

We're given two angles and an included side and we want to solve for a side opposite one of the given angles. Using the typical triangle notation we have

b=AC=91.791 meters

A=76.833 degrees

C=31.683 degrees

and our task is to find c=AB.

To apply the Law of Sines we need B, which we get since the triangle angles add to 180 degrees:

B = 180 - A - C = 180 - 76.833 - 31.683 = 71.484 degrees

Now the Law of Sines says

50.842 meters