# A triangle has corners at points A, B, and C. Side AB has a length of #42 #. The distance between the intersection of point A's angle bisector with side BC and point B is #14 #. If side AC has a length of #36 #, what is the length of side BC?

Labeling the intersection of A's bisector with side BC as D:

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This problem can be solved using the Angle Bisector Theorem, which states that the angle bisector of an angle in a triangle divides the opposite side into two segments that are proportional to the adjacent sides.

Let (D) be the point on (BC) where the angle bisector from (A) intersects (BC). Given that (AD) is the angle bisector, (AB = 42), (AC = 36), and (BD = 14), we need to find (BC).

According to the Angle Bisector Theorem,

[\frac{AB}{AC} = \frac{BD}{DC}]

Given (AB = 42), (AC = 36), and (BD = 14), let (DC = x). So,

[\frac{42}{36} = \frac{14}{x}]

Simplifying (\frac{42}{36}) gives (\frac{7}{6}), so,

[\frac{7}{6} = \frac{14}{x}]

Solving for (x),

[7x = 14 \times 6]

[x = 2 \times 6 = 12]

So, (DC = 12).

The length of (BC) is (BD + DC = 14 + 12 = 26).

Therefore, the length of side (BC) is (26).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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