Question

Two corners of a triangle have angles of `pi / 8` and `pi / 4`. 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: `~~21.05`

Explanation:

If two of the angles are `pi/8` and `pi/4`
the third angle of the triangle must be `pi - (pi/8+pi/4) = (5pi)/8`

For the longest perimeter, the shortest side must be opposite the shortest angle.
So `4` must be opposite the angle `pi/8`

By the Law of Sines
`color(white)("XXX")("side opposite "rho)/(sin(rho)) = ("side opposite " theta)/(sin(theta))` for two angles `rho` and `theta` in the same triangle.

Therefore
`color(white)("XXX")`side opposite `pi/4=(4*sin(pi/4))/(sin(pi/8)) ~~7.39`
and
`color(white)("XXX")`side opposite `(5pi)/8 = (4*sin((5pi)/8))/(sin(pi/8))~~9.66`

For a total (maximum) perimeter of
`color(white)("XXX")4+7.39+9.66= 21.05`