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 `6`. If side AC has a length of `16`, what is the length of side BC?

Answer 1

`BC=16 2/3" units"`

Explanation:

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

`rArrBC=BD+DC`

We are given BD = 6 and require to find DC

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

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

Substitute given values into this equation.

`9/16=6/(DC)`

`color(blue)"cross-multiply"`

`9xxDC=6xx16`

`rArrDC=(6xx16)/9=32/3`

`rArrBC=6+32/3=16 2/3" units"`