The hypotenuse of a right triangle is $10 \mathrm{~cm}$ long. The shorter leg is $2 \mathrm{~cm}$ shorter than the longer leg. Find the side lengths of the triangle.
Length of the shorter leg:
$\mathrm{cm}$
Length of the longer leg:
$\mathrm{cm}$
Length of the hypotenuse:
$\mathrm{cm}$

Step 1 ：Let's denote the length of the shorter leg as x, then the length of the longer leg is x + 2.

Step 2 ：According to the Pythagorean theorem, we have: \(x^2 + (x + 2)^2 = 10^2\)

Step 3 ：This is a quadratic equation that we can solve for x.

Step 4 ：The solutions to the equation are -8 and 6. However, since the length of a side of a triangle cannot be negative, we discard -8 and take 6 as the length of the shorter leg.

Step 5 ：Therefore, the length of the longer leg is 6 + 2 = 8.

Step 6 ：Final Answer: Length of the shorter leg: \(\boxed{6 \mathrm{~cm}}\), Length of the longer leg: \(\boxed{8 \mathrm{~cm}}\), Length of the hypotenuse: \(\boxed{10 \mathrm{~cm}}\)