Question

Two corners of a triangle have angles of `(2 pi )/ 3` and `( pi ) / 6`. If one side of the triangle has a length of `13`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter = 48.5167

Explanation:

`a/sin a = b/sin b = c/sin c`
The three angles are `(2pi)/3, pi/6, pi/6`

To get the longest possible perimeter, given side should correspond to the smallest angle `pi/6`

`13/sin (pi/6) = b/sin (pi/6) = c/sin ((2pi)/6)`
`b=13, c = (13 * (sin ((2pi)/3)/sin (pi/6))`
`c = (13 * sin120)/sin 60 = (13 * (sqrt3/2))/(1/2)`
`sin (pi/6) = 1/2, sin ((2pi)/3) = sin (pi/3) = sqrt3 / 2`
`c = 13*sqrt3=22.5167`

Perimeter `= 13+13+22.5167=48.5167`