Question

Two corners of a triangle have angles of `(5 pi )/ 8` and `( pi ) / 6`. If one side of the triangle has a length of `12`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter of the triangle

`color(maroon)(P = a + b + c = 48.78`

Explanation:

`hat A = (5pi)/8, hat B = pi/6, hat C = pi - (5pi)/8 - pi/6 = (5pi)/24`

To get the longest perimeter, side 12 should correspond to the least angle `hat B = pi/6`

Applying the law of Sines,

`a = (b * sin A) / sin B = (12 sin ((5pi)/8))/sin (pi/6) = 22.17`

`c = (sin C * b) / sin B = (12 * sin ((5pi)/24))/sin (pi/6) = 14.61`

Longest possible perimeter of the triangle

`color(maroon)(P = a + b + c = 22.17+ 12 + 14.61 = 48.78`