Question

A triangle has corners at points A, B, and C. Side AB has a length of `24`. 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

10

Explanation:

Here is a sketch (not to scale)

To calculate BC we require the length of DC.

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)|)))`

Substitute the appropriate values into the ratio.

hence `6/(DC)=24/16`

now cross-multiply

`rArr 24xxDC=16xx6rArr DC=(16xx6)/24=4`

Now BC = BD + DC = 6 + 4 = 10