Question

Two corners of a triangle have angles of `(5 pi ) / 12` and `(3 pi ) / 8`. If one side of the triangle has a length of `8`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter of the triangle is 32.8348

Explanation:

Given are the two angles `(5pi)/12` and `(3pi)/8` and the length 12

The remaining angle:

`= pi - (((5pi)/12) + (3pi)/8) = (5pi)/24`

I am assuming that length AB (8) is opposite the smallest angle

`a/ sin A = b / sin B = c / sin C`

`8 / sin ((5pi)/24) = b / sin ((5pi)/12) = c / sin ((3pi)/8)`

`b = (8 * sin ((5pi)/12)) / sin ((5pi)/24) = 12.6937`

`c = (8*sin ((3pi)/8)) / sin ((5pi)/24) = 12.1411`

Longest possible perimeter of the triangle is = (a + b + c) / 2 =( 8 + 12.6937 + 12.1411) = 32.8348#