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?

1 Answer
May 19, 2017

34/3 approx 11.33

Explanation:

enter image source here

Consider this image where AD is the angle bisector of A.

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

That if AD is the angle bisector of A then, color(red)[(AB)/(AC) = (BD)/(CD)], 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.

You can also watch this video.

Now, in this question,
AB = 9
AC = 8
BD = 6

=> 9/8 = 6 / (CD)

=> CD = 6*8/9 = 16/3

=> BC = BD + CD = 6 + 16/3 = (18+16)/3 = 34/3 approx 11.33