Question

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

Answer 1

`color(green)("Longest possible perimeter of the ") color(indigo)(Delta = 91.62 " units"`

Explanation:

`hat A = (5pi)/8, hat B = pi/12, hat C = pi - (5pi)/8 - pi/12 = (7pi)/24`

To find the longest possible perimeter of the triangle, we length 12 should correspond to side b as `hat B` has the least angle measure.

Applying the Law of Sines,

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

`a = (12 * sin ((5pi)/8))/ sin (pi/12) = 42.84 " units"`

`c = (12 * sin((7pi)/24)) / sin (pi/12) = 36.78 " units"`

`"Longest possible perimeter of the " Delta = (a + b + c)`

`=> 42.84 + 36.78 + 12 = 91.62 " units"`