The Angle Bisector Theorem states that in a triangle, if a line bisects one angle of a triangle and is also a median to the side opposite of that angle, then the ratio of the length of the median to the length of the side opposite the angle bisected is equal to the ratio of the length of the other two medians.
Therefore, if the angle bisector divides the opposite side into segments 6 cm and 9 cm long, the ratio of the length of the median to the length of the side opposite the angle bisected is 6/9.
Let x be the length of the third side of the triangle.
The Angle Bisector Theorem states: (x/22.5) = (6/9)
Solving for x, we get: x = (22.5 * 6/9) = 15
So the length of the third side of the triangle is 15 cm.
This is the only possible length for the third side of the triangle because the Angle Bisector Theorem only holds if the angle bisector is also a median to the side opposite of that angle.