# A triangle has corners at points A, B, and C. Side AB has a length of #15 #. 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 #24 #, what is the length of side BC?

Length of side BC = 18.2

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

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To find the length of side BC in the triangle with corners at points A, B, and C, where side AB has a length of 15, the distance between the intersection of point A's angle bisector with side BC and point B is 7, and side AC has a length of 24, we can use the angle bisector theorem.

According to the angle bisector theorem, the ratio of the lengths of the two segments formed by the intersection of an angle bisector with the opposite side in a triangle is equal to the ratio of the lengths of the other two sides of the triangle.

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

Using the angle bisector theorem, we can set up the following equation:

( \frac{AC}{AB} = \frac{BC}{BC + 7} )

Substituting the given values, we get:

( \frac{24}{15} = \frac{x}{x + 7} )

Cross multiply and solve for x:

( 24(x + 7) = 15x )

( 24x + 168 = 15x )

( 24x - 15x = -168 )

( 9x = -168 )

( x = \frac{-168}{9} )

( x = -18.67 )

Since the length of a side cannot be negative, we disregard the negative solution.

Therefore, the length of side BC is ( \boxed{18.67} ).

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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.

- A triangle has corners at points A, B, and C. Side AB has a length of #18 #. The distance between the intersection of point A's angle bisector with side BC and point B is #6 #. If side AC has a length of #14 #, what is the length of side BC?
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- Triangle A has an area of #4 # and two sides of lengths #8 # and #7 #. Triangle B is similar to triangle A and has a side with a length of #13 #. What are the maximum and minimum possible areas of triangle B?
- Triangle A has sides of lengths #1 ,3 #, and #4 #. Triangle B is similar to triangle A and has a side of length #3 #. What are the possible lengths of the other two sides of triangle B?

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