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

Answer 1

`BC=51/9=5 2/3" units"`

Explanation:

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

Then BC = BD + CD

We know that BD = 3 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)|)))`

Substituting known values into this equation.

`rArr3/(CD)=9/8`

`color(blue)"cross multiplying"`

`rArr9CD=3xx8`

`rArrCD=24/9`

`rArrBC=3+24/9=27/9+24/9=51/9=5 2/3" units"`