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The length of the hypotenuse can be found using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

So, in this case, the hypotenuse length can be found as:

hypotenuse^2 = 9^2 + 12^2
hypotenuse^2 = 81 + 144
hypotenuse^2 = 225

Taking the square root of both sides:

hypotenuse = sqrt(225)

The square root of 225 is 15.

Therefore, the length of the hypotenuse of the right triangle is 15 units.