Question

Two corners of a triangle have angles of `(3 pi ) / 8` and `pi / 3`. 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 `color(crimson)(P = 3.25`

Explanation:

`hat A = (3pi)/8, hat B = pi/3, hat C = (7pi)/24`

Least angle #hat C = (7pi)/24 should correspond to the side of length 1 to get the longest possible perimeter.

Applying the law of Sines,

`a / sin A = b / sin B = c / sin C = 1 / sin ((7pi)/24)`

`a = sin ((3pi)/8)* (1 / sin ((7pi)/24) )= 1.16`

`b = sin (pi/3) * (1/sin ((7pi)/24) )= 1.09`

Longest possible perimeter `color(crimson)(P = 1.16 + 1.09 + 1 = 3.25`