A triangle has corners at points A, B, and C. Side AB has a length of #15 #. The distance between the intersection of point A's angle bisector with side BC and point B is #12 #. If side AC has a length of #27 #, what is the length of side BC?

1 Answer
Aug 12, 2016

#bar(BC) = 33.6#

Explanation:

Applying sinus law to triangles #Delta ABD# and #Delta ADC# we have

#sin(alpha)/(bar(BD)) = sin(beta)/(bar(AB))#

and

#sin(alpha)/(bar(DC))=sin(gamma)/(bar(AC))#

but #gamma = pi - alpha# and #sin(gamma) = sin(alpha)# so the last equation reads now

#sin(alpha)/(bar(DC))=sin(beta)/(bar(AC))#

dividing the firts and last equation in both sides we have now

#bar(BD)/(bar(DC)) = bar(AB)/(bar(AC))# and

#bar(DC) = (bar(BD) bar(AC))/(bar(AB)) = (12 xx 27)/15 = 21.6#

Finally

#bar(BC) = bar(BD)+bar(DC) = 12+21.6 = 33.6#

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