Question

A triangle has two corners with angles of `( pi ) / 4` and `( 7 pi )/ 12`. If one side of the triangle has a length of `7`, what is the largest possible area of the triangle?

Answer 1

Largest possible area of the triangle is `33.467`

Explanation:

As two angles of a triangle are `pi/4` and `(7pi)/12`, the third angle is

`pi-pi/4-(7pi)/12=(12pi-3pi-7pi)/12=(2pi)/12=pi/6`

All triangles with these angles are similar. As length of one side is `7`, its area will be maximum, if this side is opposite smallest angle i.e. `pi/6`.

Area of a triangle given one side `a=7` and two angles `/_A=pi/6`, `/_B=pi/4` and `/_C=(7pi)/12` is

`(a^2sinBsinC)/(2sinA)`

= `(7^2sin((7pi)/12)sin(pi/4))/(2sin(pi/6))`

= `(49xx0.9659xx0.7071)/(2xx0.5)`

= `(49xx0.0.683)/1`

= `33.467`