# A triangle has sides A, B, and C. Sides A and B have lengths of 2 and 7, respectively. The angle between A and C is (pi)/12 and the angle between B and C is  (pi)/6. What is the area of the triangle?

##### 2 Answers
Oct 20, 2017

Kindly check whether the question is correct?

Also, it is sufficient if you know only one side, as we already have three angles known.

#### Explanation:

Kindly check whether the question is correct?
Smaller side will have smaller angle opposite to it and bigger side wall have bigger angle opposite to it.

In the given sum, side A is 2 and $\angle A i s \frac{\pi}{6} , \mathmr{and} s i \mathrm{de} B i s 7 \mathmr{and} \angle B i s \frac{\pi}{6}$.

Also, it is sufficient if you know only one side, as we already have three angles known.

Oct 20, 2017

Area of the #Delta ABC = 1120.0834

#### Explanation:

Assumption : only side B given as 7.

Three angles are $\angle A = \frac{\pi}{6} , \angle B = \frac{\pi}{12} , \angle C = \frac{3 \pi}{4}$

$\frac{A}{\sin} \left(\frac{\pi}{6}\right) = \frac{7}{\sin} \left(\frac{\pi}{12}\right) = \frac{C}{\sin} \left(\frac{3 \pi}{4}\right)$

$A = \frac{7 \cdot \sin \left(\frac{\pi}{6}\right)}{\sin} \left(\frac{\pi}{12}\right) = \ast 13.523 \ast$

$C = \frac{7 \cdot \sin \left(\frac{3 \pi}{4}\right)}{\sin} \left(\frac{\pi}{12}\right) = \ast 19.1244 \ast$

Semi perimeter $s = \frac{A + B + C}{2} = \frac{13.523 + 7 + 19.1244}{2} = \ast 19.8237 \ast$

$s - A = 19.8237 - 13.523 = 6.3007$
$s - B = 19.8237 - 7 = 12.8237$
$s - C = 19.8237 - 19.1244 = 0.6993$

Area of $\Delta A B C = \sqrt{s \left(s - A\right) \left(s - B\right) \left(s - C\right)}$
$= \sqrt{19.8237 \cdot 6.3007 \cdot 12.8237 \cdot 0.6993} = \ast 1120.0834 \ast$