# A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 8, respectively. The angle between A and C is (pi)/12 and the angle between B and C is  (5pi)/6. What is the area of the triangle?

Dec 14, 2015

No such triangle is possible.

#### Explanation:

If the angle between B and C is $\frac{5 \pi}{6}$ then it is an obtuse angle and the side opposite that angle must be longer than any other side of the triangle. Therefore A can not be less than B.

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Assuming the values for the lengths of A and B have been accidentally switched doesn't work either.

$\angle b$ (i.e. the angle between A and C) $= \frac{\pi}{12}$
and
$\angle a$ (i.e. the angle between B and C) $= \frac{5 \pi}{6}$

$\Rightarrow \angle c$ (i.e. the angle between B and C) $= \frac{\pi}{12}$
(since the interior angles of a triangle must add up to $\pi$)

$\Rightarrow$ C $=$ B $= 3$

But then we would have 2 sides of a triangle (B and C) whose length was less than the third side (A), which is clearly impossible.