The shorter leg of a right triangle is \( 9 \mathrm{~cm} \) shorter than the longer leg. The hypotenuse is \( 9 \mathrm{~cm} \) longer than the longer leg. Find the side lengths of the triangle. Length of the shorter leg: Length of the longer leg: Length of the hypotenuse: \( \mathrm{cm} \) \( \square \mathrm{cm} \) \( \square \mathrm{cm} \)

Let's call the longer leg "x". Then, according to the problem, the shorter leg is "x - 9" and the hypotenuse is "x + 9". Using the Pythagorean theorem, we know that: \begin{align*} x^2 + (x-9)^2 &= (x+9)^2 \\ x^2 + x^2 - 18x + 81 &= x^2 + 18x + 81 \\ x^2 - 36x &= 0 \\ x(x-36) &= 0 \\ \end{align*} So, either x = 0 (which doesn't make sense in this context) or x = 36. If x = 36, then the shorter leg is 27 (36-9) and the hypotenuse is 45 (36+9). Therefore, the side lengths of the triangle are: Length of the shorter leg: \boxed{27 \mathrm{~cm}} Length of the longer leg: \boxed{36 \mathrm{~cm}} Length of the hypotenuse: \boxed{45 \mathrm{~cm}}