Question

A triangle has corners at points A, B, and C. Side AB has a length of `12`. The distance between the intersection of point A's angle bisector with side BC and point B is `9`. If side AC has a length of `8`, what is the length of side BC?

Answer 1

`|BC|=15`

Explanation:

Labeling the intersection of the bisector of `/_CAB` with `BC` as `D`

The Bisector Theorem tells us that
`color(white)("XXX")abs(BD)/abs(AB)=abs(DC)/abs(AC)`

`color(white)("XXX")rArr abs(DC)=9/12xx8=6`

and since `abs(BC)=abs(BD)+abs(DC)`
`color(white)("XXX")abs(BC)=9+6=15`