Question

Two corners of a triangle have angles of `pi / 3` and `pi / 2`. If one side of the triangle has a length of `7`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter is `33.124`.

Explanation:

As two angles are `pi/2` and `pi/3`, the third angle is `pi-pi/2-pi/3=pi/6`.

This is the least angle and hence side opposite this is smallest.

As we have to find longest possible perimeter, whose one side is `7`, this side must be opposite the smallest angle i.e. `pi/6`. Let other two sides be `a` and `b`.

Hence using sine formula `7/sin(pi/6)=a/sin(pi/2)=b/sin(pi/3)`

or `7/(1/2)=a/1=b/(sqrt3/2)` or `14=a=2b/sqrt3`

Hence `a=14` and `b=14xxsqrt3/2=7xx1.732=12.124`

Hence, longest possible perimeter is `7+14+12.124=33.124`