The longer leg of a right triangle is $3 \mathrm{~cm}$ longer than the shorter leg. The hypotenuse is $6 \mathrm{~cm}$ longer than the shorter leg. Find the side lengths of the triangle. Length of the shorter leg: $\square \mathrm{cm}$ Length of the longer leg: $\quad \square \mathrm{cm}$ Length of the hypotenuse: $\square \mathrm{cm}$

Step 1 ：Let's denote the length of the shorter leg as \(x\). Then, the longer leg is \(x+3\) and the hypotenuse is \(x+6\).

Step 2 ：According to the Pythagorean theorem, we can set up the following equation: \((x+3)^2 + x^2 = (x+6)^2\).

Step 3 ：Solving this equation, we get two solutions: -3 and 9. However, the length of a side of a triangle cannot be negative, so we discard -3.

Step 4 ：Therefore, the length of the shorter leg is 9 cm.

Step 5 ：Substituting \(x=9\) into \(x+3\) and \(x+6\), we find the lengths of the longer leg and the hypotenuse.

Step 6 ：The length of the longer leg is 12 cm and the length of the hypotenuse is 15 cm.

Step 7 ：Final Answer: The length of the shorter leg is \(\boxed{9 \mathrm{~cm}}\), the length of the longer leg is \(\boxed{12 \mathrm{~cm}}\), and the length of the hypotenuse is \(\boxed{15 \mathrm{~cm}}\).