Let's call the base of the triangle "b". We can use the Pythagorean theorem to relate the base, height, and hypotenuse of the triangle:
h^2 + b^2 = c^2
where h is the height, b is the base, and c is the hypotenuse.
Substituting the given values, we get:
12^2 + b^2 = 15^2
144 + b^2 = 225
Subtracting 144 from both sides, we get:
b^2 = 81
Taking the square root of both sides, we get:
b = 9
Therefore, the base of the triangle is 9 cm.