Question

Two corners of a triangle have angles of `(7 pi )/ 12` and `pi / 8`. If one side of the triangle has a length of `1`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter of the triangle is `color(blue)(5.5972)`

Explanation:

`A = (7pi)/12, B = pi /8, C = pi - (7pi)/12 - pi / 8 = (7pi)/24`

Sideof length of 1 should correspond to smallest angle (pi/8) to get the longest perimeter.

`a / sin ((7pi)/12) = 1 / sin (pi/8) = c / sin ((7pi)/24)`

`a = (1 * sin ((7pi)/12)) / sin (pi/8) = 2.5241`

`c = (1 * sin ((7pi)/24)) / sin (pi/8) = 2.0731`

Longest possible perimeter of the triangle

`P = (a + b + c) / 2 = (2.5241 + 1 + 2.0731) / 2 = color (blue)(5.5972)`