Question

Two corners of a triangle have angles of `(3 pi ) / 4` and `pi / 12`. If one side of the triangle has a length of `7`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter of the triangle is

`color(blue)(= 39.64` units

Explanation:

`hat A = (3pi)/4, hat B = pi/12, hat C = pi - (3pi)/4 - pi/12 = pi/6`

To get the longest perimeter, side of length 7 should correspond to the least angle `pi/12`

Applying Law of Sines,

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

`a/ sin ((3pi)/4) = 7 / sin (pi/12) = c / sin (pi/6)`

`a = (7 * sin((3pi)/4))/sin(pi/12) =19.12`

`c = (7*sin(pi/6))/sin(pi/12) = 13.52`

Perimeter `= a + b+ c = 19.12+ 7 + 13.52 = 39.64`