# A triangle has corners at points A, B, and C. Side AB has a length of #21 #. The distance between the intersection of point A's angle bisector with side BC and point B is #7 #. If side AC has a length of #14 #, what is the length of side BC?

Let D be the point on BC where the angle bisector from A, intersects with BC

Then BC = BD + DC

We know BD = 7 and require to find DC.

substitute known values into the equation.

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To find the length of side BC, you can use the Angle Bisector Theorem, which states that in a triangle, the length of the side opposite a given angle is 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 given information, $AD$ bisects angle $A$.

Given that the length of side $AB$ is 21 and the length of side $AC$ is 14, we can set up the following proportion:

$\frac{BD}{DC} = \frac{AB}{AC}$

Substituting the given values, we get:

$\frac{BD}{DC} = \frac{21}{14} = \frac{3}{2}$

Now, let $x$ be the length of $BD$, so $DC$ would be $2x$ since $BD$ and $DC$ are in the ratio $3:2$.

Since the distance between the intersection of point $A$'s angle bisector with side $BC$ and point $B$ is 7, we have:

$x + 7 = 21$

Solving for $x$:

$x = 21 - 7 = 14$

Therefore, the length of side $BC$, which is $BD + DC$, is $14 + 2(14) = 14 + 28 = 42$.

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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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