Question

A triangle has corners at points A, B, and C. Side AB has a length of `45`. 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 `36`, what is the length of side BC?

Answer 1

`21 3/5" units"`

Explanation:

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

`"Then " BC= BD+DC`

`"We know " BD=12" and have to find " DC`

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

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

Substitute given values into the equation.

`rArr45/36=12/(DC)`

`color(blue)"cross multiply"`

`rArr45DC=12xx36`

`rArrDC=(12xx36)/45=48/5=9 3/5`

`rArrBC=12+9 3/5=21 3/5`