A triangle has corners at points A, B, and C. Side AB has a length of #18 #. The distance between the intersection of point A's angle bisector with side BC and point B is #3 #. If side AC has a length of #21 #, what is the length of side BC?

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Answer 1 · 2016-06-06T09:26:56

#BC=6.5#

Please refer to figure below.
enter image source here

Here let #a=BC#,
#b=AC=21# and
#c=AB=18#.

Further, bisector of angle #A# cuts #AB# at #D# and #BD=3#

In such a triangle according to angle bisector theorem, bisector of angle #A#, divides #BC#

in the ratio of the two sides containing the angle.

In other words, #(AB)/(AC)=(BD)/(DC)# and hence here we have

#18/21=3/(DC)# or #18xxDC=21xx3# and

#DC=(21xx3)/18=(7cancel(21)xx3)/(6cancel(18))=7xx3/6=7/2=3.5#

Hence #BC=3+3.5=6.5#