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 #12 #. If side AC has a length of #16 #, what is the length of side BC?

Question

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 #12  #.  If side AC has a length of #16 #, what is the length of side BC?

Allison Allen · Accepted Answer

Length of BC = 18

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #32 / 16 = 12 / (DC)# #DC = (12*cancel(16)) /( cancel(32) 2)= 6# #BC = BD+DC= 12+6 =18#

Aaliyah Anderson · Answer

undefined The length of side BC is 24.

Allison Allen · Answer

undefined To find the length of side BC, we can use the Angle Bisector Theorem, which states that in a triangle, the angle bisector of an angle divides the opposite side into segments proportional to the lengths of the other two sides.

Let $D$ be the point where the angle bisector of angle $A$ intersects side $BC$. According to the theorem, $BD:DC = AB:AC$.

Given that $AB = 32$ and $AC = 16$, we can set up the proportion:

$$BD:DC = 32:16$$

Simplifying this proportion, we get:

$$BD:DC = 2:1$$

Since $BD = 2x$ and $DC = x$ for some length $x$, we can find $x$:

$$2x:x = 12:16$$

Solving for $x$, we get:

$$2x = 12 \implies x = 6$$

Thus, $DC = 6$ and $BD = 2 \times 6 = 12$.

Therefore, the length of side $BC$ is $BD + DC = 12 + 6 = 18$.