Question

Two corners of a triangle have angles of `(2 pi )/ 3` and `( pi ) / 4`. If one side of the triangle has a length of `12`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter is `12+40.155+32.786=84.941`.

Explanation:

As two angles are `(2pi)/3` and `pi/4`, third angle is `pi-pi/8-pi/6=(12pi-8pi-3pi)/24-=pi/12`.

For longest perimeter side of length `12`, say `a`, has to be opposite smallest angle `pi/12` and then using sine formula other two sides will be

`12/(sin(pi/12))=b/(sin((2pi)/3))=c/(sin(pi/4))`

Hence `b=(12sin((2pi)/3))/(sin(pi/12))=(12xx0.866)/0.2588=40.155`

and `c=(12xxsin(pi/4))/(sin(pi/12))=(12xx0.7071)/0.2588=32.786`

Hence longest possible perimeter is `12+40.155+32.786=84.941`.