# A triangle has corners at 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 #8 #, what is the length of side BC?

Consider this image where

Triangles have a property related to angle bisectors called the angle bisector theorem :-

That if

For proofs of this theorem visit this link.

I have gone through both the proofs and found them to be correct.

You can also watch this video.

Now, in this question,

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Using the Angle Bisector Theorem and the given information:

[ \frac{AC}{AB} = \frac{AD}{DB} ]

[ \frac{8}{9} = \frac{x}{6} ]

Solving for ( x ):

[ x = \frac{8 \times 6}{9} = \frac{16}{3} ]

Now, we know that ( x + BD = BC ), where ( BD = 6 ).

[ BC = x + 6 = \frac{16}{3} + 6 = \frac{16}{3} + \frac{18}{3} = \frac{34}{3} ]

So, the length of side BC is ( \frac{34}{3} ).

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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.

- Find the value of x in the figure?
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- 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 #16 #. If side AC has a length of #32 #, what is the length of side BC?
- Triangle A has sides of lengths #36 #, #42 #, and #40 #. 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?
- A triangle has corners at points A, B, and C. Side AB has a length of #33 #. The distance between the intersection of point A's angle bisector with side BC and point B is #15 #. If side AC has a length of #27 #, what is the length of side BC?

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