Question

A triangle has corners at points A, B, and C. Side AB has a length of `42`. The distance between the intersection of point A's angle bisector with side BC and point B is `14`. If side AC has a length of `36`, what is the length of side BC?

Answer 1

`abs(BC)=26`

Explanation:

Labeling the intersection of A's bisector with side BC as D:

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

`color(white)("XXX")abs(DC)/36=14/42`

`color(white)("XXX")abs(DC)=(cancel(14)^2xxcancel(36)^6)/(cancel(42)_cancel(6))=12`

`abs(BC)=abs(BD)+abs(DC)`
`color(white)("XXX")=14+12=26`