A triangle has corners at points A, B, and C. Side AB has a length of 9 . The distance between the intersection of point A's angle bisector with side BC and point B is 6 . If side AC has a length of 8 , what is the length of side BC?

May 19, 2017

$\frac{34}{3} \approx 11.33$

Explanation:

Consider this image where $A D$ is the angle bisector of $A$.

Triangles have a property related to angle bisectors called the angle bisector theorem :-

That if $A D$ is the angle bisector of $A$ then, $\textcolor{red}{\frac{A B}{A C} = \frac{B D}{C D}}$, i.e. an angle bisector divides the opposite side in the ratio if the adjacent sides.

For proofs of this theorem visit this link.
I have gone through both the proofs and found them to be correct.

Now, in this question,
$A B = 9$
$A C = 8$
$B D = 6$

$\implies \frac{9}{8} = \frac{6}{C D}$

$\implies C D = 6 \cdot \frac{8}{9} = \frac{16}{3}$

$\implies B C = B D + C D = 6 + \frac{16}{3} = \frac{18 + 16}{3} = \frac{34}{3} \approx 11.33$