Question

A triangle has corners at points A, B, and C. Side AB has a length of `16`. The distance between the intersection of point A's angle bisector with side BC and point B is `12`. If side AC has a length of `27`, what is the length of side BC?

Answer 1

`32 1/4" units"`

Explanation:

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

`rArrBC=BD+DC`

We know BD = 12 and require to find DC

Applying the `color(blue)"Angle bisector theorem"` to the triangle.

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

Substitute known values into the equation.

`rArr16/27=12/(DC)`

`color(blue)"cross multiply"`

`rArr16DC=12xx27`

`rArrDC=(12xx27)/16=20 1/4`

`rArrBC=12+20 1/4=32 1/4" units"`