Question

A triangle has corners at points A, B, and C. Side AB has a length of `36`. The distance between the intersection of point A's angle bisector with side BC and point B is `15`. If side AC has a length of `27`, what is the length of side BC?

Answer 1

26.25

Explanation:

Here is a sketch (not to scale)

To calculate the length of BC , we require the length of DC.

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

For the triangle ABC given, this is.

`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.

`rArr15/(DC)=36/27`

now cross-multiply

`rArrDCxx36=15xx27rArrDC=(15xx27)/36`

Thus DC `=45/4=11 1/4=11.25`

length of BC = BD + DC = 15 + 11.25 = 26.25