# A triangle has sides A, B, and C. The angle between sides A and B is pi/4 and the angle between sides B and C is pi/12. If side B has a length of 24, what is the area of the triangle?

Jul 20, 2016

$= 60.86 s q u n i t$

#### Explanation:

Let the length of the perpendicular dropped on side B from opposite corner be h.And this perpendicular divides sides B(=24) in two parts.Let the length of one part towards angle $\frac{\pi}{4}$ be
x and the length of the other part towards angle $\frac{\pi}{12}$ be y.

So $\frac{x}{h} = \cot \left(\frac{\pi}{4}\right) \implies x = h$

And $\frac{y}{h} = \cot \left(\frac{\pi}{12}\right) \implies y = h \cot \left(\frac{\pi}{12}\right)$

But by the given condition
$x + y = 24$
$\implies h + h \cot \left(\frac{\pi}{12}\right) = 24$
$\therefore h = \frac{24}{1 + \cot \left(\frac{\pi}{12}\right)}$

So area of the triangle

$\text{Area} = \frac{1}{2} \cdot B \cdot h = \frac{1}{2} \times 24 \times \frac{24}{1 + \cot \left(\frac{\pi}{12}\right)}$

$= 60.86 s q u n i t$