# A triangle has corners at points A, B, and C. Side AB has a length of #10 #. The distance between the intersection of point A's angle bisector with side BC and point B is #2 #. If side AC has a length of #8 #, what is the length of side BC?

Consider the diagram

Always remember that the angle bisector of any angle in a triangle also bisects the opposite side into two equal halves

So,

We can say that

#color(purple)(bd=dc#

So,the length of

#color(green)(bc=bd+dc=2+2=4#

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To find the length of side BC in the given triangle, we can use the angle bisector theorem, which states that in a triangle, an angle bisector divides the opposite side into segments proportional to the lengths of the other two sides.

Given:

- Side AB has a length of 10.
- The distance between the intersection of point A's angle bisector with side BC and point B is 2.
- Side AC has a length of 8.

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

Using the angle bisector theorem, we can set up the proportion:

[ \frac{{AC}}{{AB}} = \frac{{\text{Length of segment } AC}}{{\text{Length of segment } CB}} ]

Plugging in the given values:

[ \frac{{8}}{{10}} = \frac{{x + 2}}{{x}} ]

Solving this equation for (x):

[ 8x = 10(x + 2) ] [ 8x = 10x + 20 ] [ 2x = 20 ] [ x = 10 ]

So, the length of side BC is (10).

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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 #3 #. The distance between the intersection of point A's angle bisector with side BC and point B is #1 #. If side AC has a length of #4 #, what is the length of side BC?
- 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 #15 #. If side AC has a length of #36 #, what is the length of side BC?
- Triangle A has an area of #24 # and two sides of lengths #8 # and #15 #. Triangle B is similar to triangle A and has a side with a length of #12 #. 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 #56 #. The distance between the intersection of point A's angle bisector with side BC and point B is #12 #. If side AC has a length of #42 #, what is the length of side BC?
- Please solve q 60 ?

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