Question

A triangle has corners at points A, B, and C. Side AB has a length of `16`. 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 `18`, what is the length of side BC?

Answer 1

`BC=19.125`

Explanation:

Let us draw the figure as given below:

Here `AP` bisects `/_A` and intersects `BC` at `P`. As the distance between the intersection of point `/_A`'s angle bisector with side `BC` and point `B` is `9`. Hence while `BP=9`, `AB=16` and `AC=18`, as already given.

Now as `AP` bisects `/_A`, from angle bisector theorem, we have

`(AC)/(AB)=(CP)/(PB)` and hence

`18/16=(CP)/9` and therefore

`CP=(18xx9)/16=(cancel18^9xx9)/(cancel16^8)=81/8=10.125`

Hence `BC=BP+CP=10.125+9=19.125`