A triangle has corners points A, B, and C. Side AB has a length of #9 #. The distance between the intersection of point A's angle bisector with side BC and point B is #6 #. If side AC has a length of #14 #, what is the length of side BC?

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Answer 1 · 2017-06-10T19:51:09

The length of #BC=46/3=15.3#

Let #X# be the point of intersection of the bisector with #BC#

We apply the sine rule to triangle #ABX#

#(AB)/(sin hat(AXB))=(BX)/sin hat(BAX)#

#9/(sin hat(AXB))=6/sin hat(BAX)#.......#(1)#

Then, we apply the sine rule to triangle #ACX#

#(AC)/(sin hat(AXC))=(XC)/sin hat(CAX)#

#(14)/(sin hat(AXC))=(XC)/sin hat(CAX)#.........#(2)#

Combining equations #(1)# and #(2)#

#hat(BAX)=hat(CAX)#

And

#sin hat(AXB)=sin hat(AXC)# as they are supplementary angles

#sin(pi-theta)=sin theta#

#9/6=14/(XC)#

#XC=(14*6)/9=28/3#

Therefore,

#BC=BX+XC=6+28/3=46/3=15.3#