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 `4`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter = 14.928

Explanation:

Sum of the angles of a triangle `=pi`

Two angles are `(2pi)/3, pi/6`
Hence `3^(rd)`angle is `pi - ((2pi)/3 + pi/6) = pi/6`

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

To get the longest perimeter, length 2 must be opposite to angle `pi/24`

`:. 4/ sin(pi/6) = b/ sin((pi)/6) = c / sin ((2pi)/3)`

`b = (4 sin((pi)/6))/sin (pi/6) = 4`

`c =( 4* sin((2pi)/3))/ sin (pi/6) = 6.9282`

Hence perimeter `= a + b + c = 4 + 4 + 6.9282 = 14.9282`