# A triangle has corners at points A, B, and C. Side AB has a length of 24 . The distance between the intersection of point A's angle bisector with side BC and point B is 4 . If side AC has a length of 28 , what is the length of side BC?

Oct 26, 2016

$\frac{28}{24} \cdot 4 = \frac{14}{3}$

#### Explanation:

Call $\alpha = \frac{\hat{A}}{2}$
Use the sin theorem twice on the 2 little triangles identified by bisector (call $\beta$ and $\pi - \beta$ the angles opposite to bar(AC and $\overline{A B}$ in these triangles and remember that $\sin \left(\beta\right) = \sin \left(\pi - \beta\right)$)

$\frac{x}{\sin} \alpha = \frac{28}{\sin} \beta$

and

$\frac{4}{\sin} \alpha = \frac{24}{\sin} \beta$

substitute and obtain $x = \frac{28}{24} \cdot 4 = \frac{14}{3}$