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 #36 #, what is the length of side BC?

Question

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 #36 #, what is the length of side BC?

Caroline Aucoin · Accepted Answer

Length of side BC = 40.8

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #15 / 36 = 12 / (DC)# #DC = (12*36) / 15 = 28.8# #BC = BD+DC= 12+28.8 =40.8#

James Atkins · Answer

40.8

I may have misinterpreted the question, but using the wonderful power of MS Paint:

Let D be the point at which the bisector of A meets BC.
Let #BC=x#
#:. DC=x-12#
Let #BhatAC=2theta#
#:.BhatAD=DhatAC#

Let #BhatDA=alpha#
(I know I drew it wrong on the diagram, but) #:.AhatDC=180-alpha# (even though I wrote #alpha#. Just ignore that bit...)

from the sine rule in #triangleABD#:

#sintheta/12=sinalpha/15#
#sinalpha=(15sintheta)/12#

from #triangleACD#:

#sintheta/(x-12)=sin(180-alpha)/36#
#sin(180-alpha)=(36sintheta)/(x-12)#

From the graph #y=sinx# or from looking at the unit circle, #sinalpha=sin(180-alpha)#

#sinalpha=(36sintheta)/(x-12)#

But #sinalpha=(15sintheta)/12#

#(36cancelsintheta)/(x-12)=(15cancelsintheta)/12#

#36/(x-12)=(5cancel15)/(4cancel12)#

#5(x-12)=36*4#
#5x-60=144#
#5x=204#
#x=40.8#

Sophia Alfred · Answer

undefined Using the Angle Bisector Theorem, we can determine the length of side BC.

The Angle Bisector Theorem states that in a triangle, an angle bisector divides the opposite side into segments that are proportional to the lengths of the other two sides.

Let D be the point where the angle bisector from A intersects side BC.

According to the theorem:
BD/DC = AB/AC

Substitute the given values:
12/(BC - 12) = 15/36

Cross multiply and solve for BC:
12 * 36 = 15 * (BC - 12)
432 = 15BC - 180
15BC = 612
BC = 612/15
BC = 40.8

Therefore, the length of side BC is approximately 40.8.