A triangle has corners at points A, B, and C. Side AB has a length of #44 #. The distance between the intersection of point A's angle bisector with side BC and point B is #16 #. If side AC has a length of #32 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #44  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #16  #.  If side AC has a length of #32 #, what is the length of side BC?

Aaliyah Anderson · Accepted Answer

≈ 27.64

The first step is to let the point where the angle bisector intersects with BC be D.

Then by the #color(blue)" Angle bisector theorem " #
# (BD)/(DC) = (AB)/(AC) #
Require to find DC.

Substitute the appropriate values into the ratio to obtain.

# 16/(DC) = 44/32 #
Now cross-multiply

#rArr 44xxDC = 32xx16 #
now divide both sides by 44

# (cancel(44) DC)/cancel(44) = (32xx16)/44 #
#rArr DC ≈ 11.64 #
and BC = BD + DC = 16 + 11-64 = 27.64

Maverick Smith · Answer

undefined Using the Angle Bisector Theorem, we find that $ \frac{AC}{AB} = \frac{CD}{BD} $, where $ D $ is the point where the angle bisector of $ \angle A $ intersects side $ BC $.

Given $ AC = 32 $ and $ AB = 44 $, we can solve for $ BD $.

$$ \frac{32}{44} = \frac{CD}{BD} $$
$$ \frac{8}{11} = \frac{CD}{BD} $$

Given that $ CD = 16 + 16 = 32 $ (since it's the distance from the angle bisector to $ B $), we can solve for $ BD $.

$$ \frac{8}{11} = \frac{32}{BD} $$
$$ BD = \frac{11 \times 32}{8} $$
$$ BD = 44 $$

So, $ BC = BD + CD = 44 + 32 = 76 $.