Question

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?

Answer 1

`34/3 approx 11.33`

Explanation:

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.](https://en.wikipedia.org/wiki/Angle_bisector_theorem#Proofs)
I have gone through both the proofs and found them to be correct.

[You can also watch this video.](https://www.khanacademy.org/math/geometry/hs-geo-similarity/hs-geo-angle-bisector-theorem/v/angle-bisector-theorem-proof)

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`