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 `6`. If side AC has a length of `8`, what is the length of side BC?

Answer 1

`BC=10" units"`

Explanation:

Let D be the point on BC where the angle bisector, intersects with BC

`"Then "BC=BD+CD`

We know that BD = 6 and have to find CD

Using the `color(blue)"angle bisector theorem"`

`color(red)(bar(ul(|color(white)(2/2)color(black)((BD)/(CD)=(AB)/(AC))color(white)(2/2)|)))`

Substitute known values into this equation and solve for CD

`rArr6/(CD)=12/8`

`color(blue)"cross multiplying"`

`rArr12CD=6xx8`

`rArrCD=48/12=4`

`rArrBC=6+4=10" units"`