A triangle has corners at points A, B, and C. Side AB has a length of #48 #. 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 #36 #, what is the length of side BC?

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Answer 1 · 2016-11-10T15:10:25

#63/4#

Call #P# the intersection point in the question.
Then using the sin theorem applied on triangles APC and APB we get

#36/sin(AhatPC)=bar(PC)/sin(PhatAC)#

#48/sin(pi-AhatPC)=9/sin(PhatAB)#

#PhatAC=PhatAB\ \ \ # by consdtruction and

#sin(AhatPC)=sin(pi-AhatPC)#

#bar(PC)/36=sin(PhatAC)/sin(AhatPC)=sin(PhatAB)/sin(pi-AhatPC)=9/48#

so

#bar(PC)=9*36/48=9*3/4#

and

#bar(BC)=bar(BP)+bar(PC)=9+27/4=(36+27)/4=63/4#