# A triangle has sides A, B, and C. Sides A and B have lengths of 2 and 1, respectively. The angle between A and C is (5pi)/24 and the angle between B and C is  (5pi)/24. What is the area of the triangle?

Nov 17, 2017

$3.20 \cdot {10}^{- 2}$

#### Explanation:

$\text{Area of a triangle} = \frac{1}{2} a b \sin C$

$a$ and $b$ are already known as $2$ and $2$, so $\frac{1}{2} \cdot 2 \cdot 1 = 1$

$\sin C$ is less obvious. $\angle a b = C$

$\angle a c = \frac{5 \pi}{24}$ and $\angle b c = \frac{5 \pi}{24}$

$\setminus \Sigma \angle = \angle a b + \angle b c + \angle a c = \pi$

$\angle a b = \pi - \angle a c - \angle b c = \pi - 2 \left(\frac{5 \pi}{24}\right) = \frac{7 \pi}{12}$

$\text{Area of the triangle} = \sin C = \sin \left(\frac{7 \pi}{12}\right) = 3.20 \cdot {10}^{- 2}$

Feb 17, 2018

Triangle cannot exist with the given information.

#### Explanation:

Given : $a = 2 , \hat{A} = \frac{5 \pi}{24} , b = 1 , \hat{B} = \frac{5 \pi}{24}$

Though the two angles are equal, sides are not.

In any triangle, the largest side and largest angle are opposite one another. In any triangle, the smallest side and smallest angle are opposite one another. ... Alternately, if two angles are congruent (equal in measure), then the corresponding two sides will be congruent (equal in measure).

Since the above condition is not satisfied, Triangle given in the sucasum cannot exist.