# A triangle has sides A, B, and C. The angle between sides A and B is (3pi)/4 and the angle between sides B and C is pi/12. If side B has a length of 22, what is the area of the triangle?

Mar 7, 2018

Area of the $\Delta A B C$ color(indigo)(A_t = 88.5781 sq units

#### Explanation:

Third angle $\hat{B} = \pi - \hat{A} - \hat{C} = \pi - \frac{\pi}{12} - \frac{3 \pi}{4} = \frac{\pi}{6}$

$\frac{22}{\sin} \left(\frac{\pi}{6}\right) = \frac{a}{\sin} \left(\frac{\pi}{12}\right) = \frac{c}{\sin} \left(\frac{3 \pi}{4}\right)$

$c = \frac{22 \cdot \sin \left(\frac{3 \pi}{4}\right)}{\sin} \left(\frac{\pi}{6}\right) = \frac{22 \cdot \left(\frac{1}{\sqrt{2}}\right)}{\frac{1}{2}} = 22 \sqrt{2}$

Using SAS formula to calculate the area of the triangle,

${A}_{t} = \left(\frac{1}{2}\right) b c \sin A = \left(\frac{1}{2}\right) \cdot 22 \cdot 22 \sqrt{2} \cdot \sin \left(\frac{\pi}{12}\right)$

color(indigo)(A_t = 88.5781 sq units