A triangle has corners at points A, B, and C. Side AB has a length of #12 #. The distance between the intersection of point A's angle bisector with side BC and point B is #9 #. If side AC has a length of #11 #, what is the length of side BC?

1 Answer
Apr 23, 2016

17.25

Explanation:

The first step is to let D be the point on BC where the angle bisector intersects with it.

Then 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)|)))#
Require to find DC

Substitute the appropriate values into the ratio to obtain

# 9/(DC) = 12/11 #

and cross-multiplying: #12xxDC = 11xx9 #

To obtain DC ,divide both sides by 12

# (cancel(12) DC)/cancel(12) = (11xx9)/12 #

#rArr DC = 8.25 #

Now, BC = BD + DC = 9 + 8.25 = 17.25