Question

Two corners of a triangle have angles of `(5 pi )/ 12` and `( pi ) / 3`. If one side of the triangle has a length of `9`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter = 32.3169

Explanation:

Sum of the angles of a triangle `=pi`

Two angles are `(5pi)/12, pi/3`
Hence `3^(rd)`angle is `pi - ((5pi)/12 + pi/3) = pi/4`

We know`a/sin a = b/sin b = c/sin c`

To get the longest perimeter, length 2 must be opposite to angle `pi/4`

`:. 9/ sin(pi/4) = b/ sin((5pi)/12) = c / sin ((pi)/3)`

`b = (9 sin((5pi)/12))/sin (pi/4) = 12.2942`

`c =( 9* sin((pi)/3))/ sin (pi/4) = 11.0227`

Hence perimeter `= a + b + c = 9 + 12.2942 + 11.0227 = 32.3169`