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 #12 #. If side AC has a length of #42 #, 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 #12  #.  If side AC has a length of #42 #, what is the length of side BC?

Allison Allen · Accepted Answer

Length of side BC = 26

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 / 42 = 12 / (DC)# #DC = (12*42) / 36 = 14# #BC = BD+DC= 12+14 =26#

Sadie Allen · Answer

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

Let's denote the length of BC as $ x $.

According to the Angle Bisector Theorem:

$$
\frac{{AC}}{{AB}} = \frac{{BC}}{{AB}}
$$

Given that AC = 42 and AB = 36:

$$
\frac{{42}}{{36}} = \frac{{x}}{{36}}
$$

Now, let's solve for $ x $:

$$
\frac{{42}}{{36}} = \frac{{7}}{{6}} = \frac{{x}}{{36}}
$$

Multiply both sides by 36:

$$
x = \frac{{7}}{{6}} \times 36 = 42
$$

So, the length of side BC is 42.