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 `6`, what is the largest possible area of the triangle?

Answer 1

Largest possible Area of triangle `color(green)(A_t = 24.6`

Explanation:

`hat A = pi/4, hat B = (7pi)/12, hat C = pi - pi/4 - (7pi)/12 = pi/6`

To get largest area, side of length 6 should correspond to least angle `hat hat C = pi/6`

As per Law of Sines, `a / sin A = c / sin C`

`a = (6 * sin (pi/4)) / sin (pi/6) = 8.49`

Area of triangle `A_t = (1/2) a c sin B`

`A_t = (1/2) * 8.49 * 6 * sin ((7pi)/12)`

Largest possible area of triangle `color(green)(A_t = 24.6`