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 #3 #. If side AC has a length of #8 #, what is the length of side BC?

Let D be the point on BC where the angle bisector, intersects BC

Then BC = BD + CD

We know that BD = 3 and have to find CD

Substituting known values into this equation.

To find the length of side BC in the triangle with corners at points A, B, and C, we can use the angle bisector theorem.

According to the angle bisector theorem, in a triangle, if an angle bisector intersects the opposite side, then it divides that side into segments that are proportional to the lengths of the other two sides of the triangle.

Let D be the point where the angle bisector of angle A intersects side BC. Then, BD/DC = AB/AC.

Given that AB = 9, AC = 8, and BD = 3, we can set up the proportion:

3/DC = 9/8

Now, we solve for DC:

3/DC = 9/8

Cross multiplying:

8 * 3 = 9 * DC

24 = 9 * DC

Divide both sides by 9:

DC = 24/9

DC = 8/3

Since the length of side AC is 8, and we found that DC = 8/3, then BC = BD + DC = 3 + 8/3.

To add the fractions, we first find a common denominator:

BC = 3 + 8/3 = 9/3 + 8/3 = (9 + 8)/3 = 17/3.

Therefore, the length of side BC is 17/3 units.

To find the length of side BC in the triangle, we can use the Angle Bisector Theorem. According to this theorem, in a triangle, an angle bisector divides the opposite side into segments that are proportional to 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 proportion:

[\frac{AC}{AB} = \frac{BC}{BA}]

Given that AC = 8 and AB = 9, we can plug in these values:

[\frac{8}{9} = \frac{x}{3}]

Now, cross-multiply to solve for x:

[8 \times 3 = 9x]

[24 = 9x]

[x = \frac{24}{9}]

[x = \frac{8}{3}]

So, the length of side BC is ( \frac{8}{3} ) or approximately 2.67.