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 `9`. If side AC has a length of `11`, what is the length of side BC?

Answer 1

17.25

Explanation:

The first step is to let D be the point on BC where the angle bisector intersects with it.

Then using the `color(blue)" Angle bisector theorem "`

`color(red)(|bar(ul(color(white)(a/a)color(black)( (BD)/(DC) = (AB)/(AC))color(white)(a/a)|)))`
Require to find DC

Substitute the appropriate values into the ratio to obtain

`9/(DC) = 12/11`

and cross-multiplying: `12xxDC = 11xx9`

To obtain DC ,divide both sides by 12

`(cancel(12) DC)/cancel(12) = (11xx9)/12`

`rArr DC = 8.25`

Now, BC = BD + DC = 9 + 8.25 = 17.25