Question

Two corners of a triangle have angles of `(5 pi )/ 8` and `( pi ) / 12`. If one side of the triangle has a length of `3`, what is the longest possible perimeter of the triangle?

Answer 1

The maximum perimeter is 22.9

Explanation:

The maximum perimeter is achieved, when you associate the given side with the smallest angle.

Calculate the third angle:
`(24pi)/24 - (15pi)/24 - (2pi)/24 = (7pi)/24`

`pi/12` is the smallest

Let angle `A = pi/12` and the length of side `a = 3`
Let angle `B = (7pi)/24`. The length of side b is unknown
Let angle `C = (5pi)/8`. The length of side c is unknown.

Using the law of sines:

The length of side b:

`b = 3sin((7pi)/24)/sin(pi/12) ~~ 9.2`

The length of side c:

`c = 3sin((5pi)/8)/sin(pi/12) ~~ 10.7`

P = 3 + 9.2 + 10.7 = 22.9