A triangle has corners at points A, B, and C. Side AB has a length of #32 #. The distance between the intersection of point A's angle bisector with side BC and point B is #4 #. If side AC has a length of #28 #, what is the length of side BC?

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Answer 1 · 2016-08-05T20:49:06

#norm(B-C)= 7.5#

Referencing the attached figure, we have two triangles

#ABD# and #ADC#. Aplying the sinus law we have

#ABD-> sin(alpha)/4=sin(beta)/32#
#ADC->sin(alpha)/x = sin(gamma)/28#

but #beta = pi-gamma# so the first equation reads

#ABD-> sin(alpha)/4=sin(gamma)/32# because #sin(pi-gamma)=sin(gamma)#

Dividing term to term the two equations we have

#4/x=32/28# giving #x = 7/2# so #norm(B-C)=4+7/2 = 7.5#

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