A triangle has corners at points A, B, and C. Side AB has a length of #36 #. The distance between the intersection of point A's angle bisector with side BC and point B is #18 #. If side AC has a length of #27 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #36  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #18  #.  If side AC has a length of #27 #, what is the length of side BC?

Ella Armstrong · Accepted Answer

Length of side BC = 31.5

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #36 / 27 = 18 / (DC)# #DC = (18*27) / 36 = 13.5# #BC = BD+DC= 18+13.5 =31.5#

Wyatt Anderson · Answer

undefined Using the Angle Bisector Theorem, we can find the length of side BC. The theorem states that in a triangle, the length of a side is proportional to the lengths of the other two sides that form the angle bisector. Let the intersection point of the angle bisector of angle A with side BC be D. According to the theorem:

$$ \frac{BD}{DC} = \frac{AB}{AC} $$

Given that AB = 36 and AC = 27:

$$ \frac{BD}{DC} = \frac{36}{27} = \frac{4}{3} $$

Since BD = 18 (given in the question), we can solve for DC:

$$ \frac{18}{DC} = \frac{4}{3} $$
$$ DC = \frac{3}{4} \times 18 = 13.5 $$

Therefore, BC = BD + DC = 18 + 13.5 = 31.5. Thus, the length of side BC is 31.5.