A triangle has sides A, B, and C. Sides A and B are of lengths 5 and 7, respectively, and the angle between A and B is (3pi)/4 . What is the length of side C?

Feb 9, 2016

$C = \sqrt{74 + 35 \sqrt{2}} \approx 11.11$

Explanation:

Let the angle between $A$ and $B$ be $\gamma = \frac{3 \pi}{4}$.

Then, the length of side $C$ can be computed with the law of cosines:

${C}^{2} = {A}^{2} + {B}^{2} - 2 A B \cdot \cos \left(\gamma\right)$

$= {5}^{2} + {7}^{2} - 2 \cdot 5 \cdot 7 \cdot \cos \left(\frac{3 \pi}{4}\right)$

$= 74 - 70 \cos \left(\frac{3 \pi}{4}\right)$

$= 74 - 70 \cdot \left(- \cos \left(\frac{\pi}{4}\right)\right)$

$= 74 + 70 \cdot \frac{\sqrt{2}}{2}$

$= 74 + 35 \sqrt{2}$

Thus,

$C = \sqrt{74 + 35 \sqrt{2}} \approx 11.11$