# 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?

Length of side BC = 26

Let the point where the angle bisector intersects with side BC be D

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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.

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