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

Answer 1

Longest possible perimeter is `137.434`

Explanation:

As two angles are `(5pi)/8` and `pi/12`, third angle is

`pi-(5pi)/8-pi/12=(24pi)/24-(15pi)/24-(2pi)/24=(7pi)/24`

the smallest of these angles is `pi/12`

Hence, for longest possible perimeter of the triangle, the side with length `18`, will be opposite the angle `pi/12`.

Now for other two sides, say `b` and `c`, we can use sine formula, and using it

`18/sin(pi/12)=b/sin((5pi)/8)=c/sin((7pi)/24)`

or `18/0.2588=b/0.9239=c/0.7933`

therefore `b=(18xx0.9239)/0.2588=64.259`

and `c=(18xx0.7933)/0.2588=55.175`

and perimeter is `64.259+55.175+18=137.434`