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 #9 #. If side AC has a length of #18 #, what is the length of side BC?

1 Answer
Jan 9, 2017

#BC=19.125#

Explanation:

Let us draw the figure as given below:
enter image source here
Here #AP# bisects #/_A# and intersects #BC# at #P#. As the distance between the intersection of point #/_A#'s angle bisector with side #BC# and point #B# is #9#. Hence while #BP=9#, #AB=16# and #AC=18#, as already given.

Now as #AP# bisects #/_A#, from angle bisector theorem, we have

#(AC)/(AB)=(CP)/(PB)# and hence

#18/16=(CP)/9# and therefore

#CP=(18xx9)/16=(cancel18^9xx9)/(cancel16^8)=81/8=10.125#

Hence #BC=BP+CP=10.125+9=19.125#