# A triangle has corners at points A, B, and C. Side AB has a length of #45 #. 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 #42 #, what is the length of side BC?

By the angle bisector theorem, I get

We call the corners "vertices" -- it sounds smarter.

Let's call the foot of A's angle bisector D. That's where it meets BC.

By the angle bisector theorem, we have proportionality

#{AC}/{CD} = {AB}/{BD}

# CD = frac{AC \ BD}{AB} = { (42)(6) }/45 = 28/5

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Using the angle bisector theorem, we can find the length of side BC. Let D be the point where the angle bisector of angle A intersects side BC.

According to the angle bisector theorem: [ \frac{BD}{DC} = \frac{AB}{AC} ]

Given: AB = 45, AC = 42, and BD = 6. [ \frac{6}{DC} = \frac{45}{42} ] [ \frac{6}{DC} = \frac{15}{14} ] [ 6 \times 14 = 15 \times DC ] [ 84 = 15 \times DC ] [ DC = \frac{84}{15} ] [ DC = 5.6 ]

Now, we can find the length of BC using the fact that BC = BD + DC. [ BC = 6 + 5.6 ] [ BC = 11.6 ]

Therefore, the length of side BC is 11.6.

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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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