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 `9`. If side AC has a length of `45`, what is the length of side BC?

Answer 1

`BC=18 9/14" units" ≈18.643" to 3 dec. places"`

Explanation:

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

Then BC = BD + CD

We know that BD = 9 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.

`rArr9/(CD)=42/45`

`color(blue)"cross multiplying"`

`rArr42CD=9xx45`

`rArrCD=(9xx45)/42=(cancel(9)^3xx45)/cancel(42)^(14)=135/14=9 9/14`

`rArrBC=9+9 9/14=18 9/14≈18.643" to 3 dec.places"`