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

Answer 1

The perimeter is 61

Explanation:

Let `/_A = pi/12`
Let `/_B = (5pi)/8`
Then `/_C = pi - (5pi)/8 - pi/12 = (24pi)/24 - (15pi)/24 - (2pi)/24 = (7pi)/24`

Let side `a` be the side opposite `/_A` with length `8`. We do this because associating the given side with the smallest angle will yeild the largest perimeter.

Using the Law of Sines we can write the equation for sides `b` and `c`:

`b = asin(/_B)/sin(/_A)`
`c = asin(/_C)/sin(/_A)`

Let p = the perimeter

`p = a + a(sin(/_B) + sin(/_C))/sin(/_A)`

`p = 61`