# A triangle has corners at points A, B, and C. Side AB has a length of #7 #. 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?

Given AB = 7, AC = 14 and BD = 6.

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Using the Angle Bisector Theorem, the length of BC can be calculated as (BC = \frac{AB \times AC}{AB + AC}). Substituting the given values, we get (BC = \frac{7 \times 14}{7 + 14}), which simplifies to (BC = \frac{98}{21}), or approximately (BC = 4.67). Therefore, the length of side BC is approximately 4.67.

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Using the angle bisector theorem, we can determine the length of side BC. According to the theorem, in a triangle, an angle bisector divides the opposite side into segments that are proportional to the lengths of the adjacent sides.

Let the point where the angle bisector of angle A intersects side BC be point D. Since AD is the angle bisector of angle A, we can set up the following proportion:

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

Plugging in the values we have:

[ \frac{BD}{DC} = \frac{7}{14} ]

[ \frac{BD}{DC} = \frac{1}{2} ]

Now, we know that BD = 6 (the distance between the intersection of the angle bisector with side BC and point B). We can set up another proportion using this information:

[ \frac{BD}{DC} = \frac{6}{x} ]

Where x is the length of side BC. Plugging in the values we have:

[ \frac{6}{x} = \frac{1}{2} ]

Solving for x:

[ x = \frac{6}{\frac{1}{2}} ]

[ x = 6 \times 2 ]

[ x = 12 ]

Therefore, the length of side BC is 12.

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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 points A, B, and C. Side AB has a length of #9 #. 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?
- Triangle A has an area of #60 # and two sides of lengths #12 # and #15 #. Triangle B is similar to triangle A and has a side of length #5 #. What are the maximum and minimum possible areas of triangle B?
- Two sides of an isosceles triangle have lengths of 3 and 6. What could be the length of the third side?
- Triangle A has an area of #6 # and two sides of lengths #3 # and #8 #. Triangle B is similar to triangle A and has a side with a length of #7 #. What are the maximum and minimum possible areas of triangle B?
- A triangle has corners at points A, B, and C. Side AB has a length of #27 #. The distance between the intersection of point A's angle bisector with side BC and point B is #18 #. If side AC has a length of #36 #, what is the length of side BC?

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