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 #9 #. If side AC has a length of #36 #, what is the length of side BC?

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

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

Let's denote the length of side BC as ( x ). According to the theorem, the ratio of the length of side AB to the length of side AC is equal to the ratio of the length of the segment BC to the length of the segment BD, where D is the point where the angle bisector of angle A intersects side AC.

So, we have:

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

Given that AB = 45, AC = 36, and BD = 9 (given in the problem), we can substitute these values into the equation:

[\frac{45}{36} = \frac{x}{9}]

Solving for ( x ), we get:

[x = \frac{45 \times 9}{36}]

[x = \frac{405}{36}]

[x = 11.25]

Therefore, the length of side BC is 11.25 units.

To find the length of side BC, we can use the angle bisector theorem. According to this theorem, in a triangle, the ratio of the lengths of the two segments of a side bisected by an angle bisector is equal to the ratio of 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. According to the angle bisector theorem:

[\frac{BD}{DC} = \frac{AB}{AC}]

Given that AB = 45, AC = 36, and BD = 9 (since the distance between the intersection of the angle bisector with side BC and point B is 9), we can solve for DC:

[\frac{9}{DC} = \frac{45}{36}]

Solving for DC:

[DC = \frac{36 \times 9}{45} = 7.2]

Now, BC = BD + DC:

[BC = 9 + 7.2 = 16.2]

So, the length of side BC is 16.2 units.