Question

Two corners of a triangle have angles of `pi / 3` and `pi / 6`. If one side of the triangle has a length of `1`, what is the longest possible perimeter of the triangle?

Answer 1

Largest possible perimeter of the triangle is 4.7321

Explanation:

Sum of the angles of a triangle `=pi`

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

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

To get the longest perimeter, length 2 must be opposite to angle `pi/6`

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

`b = (1*sin(pi/3))/sin (pi/6) = 1.7321`

`c =( 1* sin(pi/2)) /sin (pi/6) = 2`

Hence perimeter `= a + b + c = 1 + 1.7321 + 2 = 4.7321`