A triangle has corners at points A, B, and C. Side AB has a length of #24 #. The distance between the intersection of point A's angle bisector with side BC and point B is #9 #. If side AC has a length of #35 #, what is the length of side BC?

Let the point where the angle bisector intersects with side BC be D

Using the Angle Bisector Theorem, we can find the length of side BC.

The Angle Bisector Theorem states that in a triangle, if an angle bisector divides the opposite side into segments, then those segments are proportional to the lengths of the other two sides of the triangle.

Let D be the point where the angle bisector from A intersects side BC. According to the problem, AD is the angle bisector of angle A.

Now, we have:

(\frac{BD}{DC} = \frac{AB}{AC})

Plugging in the given values:

(\frac{BD}{DC} = \frac{24}{35})

Given that the distance between D and B is 9, we can express BD as BC - 9. So, we have:

(\frac{BC - 9}{DC} = \frac{24}{35})

We also know that BC = BD + DC. Substituting BD = BC - 9, we get:

(\frac{BC - 9}{DC} = \frac{24}{35})

Now, we need to find DC. We know that DC + BD = BC, and BD = BC - 9. So, DC = 9.

Substituting DC = 9 into the equation:

(\frac{BC - 9}{9} = \frac{24}{35})

Now, cross-multiplying:

(35(BC - 9) = 24 \times 9)

(35BC - 315 = 216)

(35BC = 531)

(BC = \frac{531}{35})

(BC = 15.17) (approximately)

So, the length of side BC is approximately 15.17 units.