Algebra question

Question

In triangle the length b=33 and the measure of an angle B = 120° are known. Find the area of the triangle if the sum of lengths of the rest two sides is 36.

Answer 1

There are probably better ways to solve this problem, but this works. Let one of the other two sides have length x. Then, using Cartesian coordinates, the coordinates of the endpoints of the side of length x can be (0, 0), (x, 0). Then, given the 120° angle, the coordinates of the side of length y will be (x, 0) and (x + 1/2y, √3/2 y) , where y = 36 - x, so the endpoint is (x + 1/2(36-x),√3/2(36 - x)) = (18 + 1/2x, √3/2(36 - x))

Then, as the difference between (0, 0) and (18 + 1/2x, √3/2(36 - x)) is of length 33,

324 + 18x + 1/4x^2 + 3/4(1296 - 72x + x^2) = 33^2= 1089

1296 - 36x + x^2 = 1089

x^2 - 36x + 207 = 0

(x - 18 - 3√13)(x - 18 + 3√13) = 0

Then, the area of the triangle is 1/2 xy sin 120° = 1/2(18 + 3 √13)(18 - 3 √13)√ 3 / 2 =

1/2(207)√ 3 / 2 = 207√ 3/4 = 89.6336