Question

Two corners of a triangle have angles of `(7 pi )/ 12` and `pi / 4`. 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

`color(blue)(P + a + b + c ~~ 34.7685`

Explanation:

`hatA = (7pi)/12, hatB = pi/4, side = 8`

To find the longest possible perimeter of the triangle.

Third angle `hatC = pi - (7pi)/12 - pi/4 = pi/6`

To get the longest perimeter, smallest angle `hatC = pi/6` should correspond to side length 8#

Using sine law, `a / sin A = b / sin B = c / sin C`

`a = (c * sin A) / sin C = (8 * sin((7pi)/12)) / sin (pi/6) = 15.4548`

`b = (c * sin B) / sin C = (8 * sin(pi/4) )/ sin (pi/6) = 11.3137`

Longest possible perimeter of the triangle is

`color(blue)(P + a + b + c = 15.4548 + 11.3137 + 8 = 34.7685`