A triangle has corners at points A, B, and C. Side AB has a length of #36 #. The distance between the intersection of point A's angle bisector with side BC and point B is #14 #. If side AC has a length of #36 #, what is the length of side BC?

To find the length of side BC, 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 other two sides of the triangle.

Given that side AB has a length of 36 and side AC has a length of 36, and the distance between the intersection of point A's angle bisector with side BC and point B is 14, we can set up the following proportion:

$\frac{BC}{AC} = \frac{BD}{AD}$

Where:

• BC is the length of side BC.
• AC is the length of side AC.
• BD is the distance between the intersection of point A's angle bisector with side BC and point B.
• AD is the length of the remaining portion of side AC.

We know that AC = 36 and BD = 14.

To find AD, we subtract BD from AC: $AD = AC - BD = 36 - 14 = 22$

Now we can solve for BC: $\frac{BC}{36} = \frac{14}{22}$

Cross-multiplying: $22 \cdot BC = 14 \cdot 36$ $22 \cdot BC = 504$

Dividing both sides by 22: $BC = \frac{504}{22}$ $BC = 22.9$

So, the length of side BC is approximately 22.9.