The shortest leg of a triangle is 1 inches shorter than the other leg. The hypotenuse of this triangle is 5 inches. What are the lengths of the two legs of this triangle?

The shortest leg is inches long.

The other leg is inches long.

Step 1 ：Given that the shortest leg of a triangle is 1 inch shorter than the other leg and the hypotenuse of this triangle is 5 inches, we need to find the lengths of the two legs of the triangle.

Step 2 ：We can set up the equation as follows: \((a-1)^2 + a^2 = 5^2\). This simplifies to \(2a^2 - 2a - 24 = 0\).

Step 3 ：We can solve this quadratic equation for a using the quadratic formula: \(a = [-(-2) ± \sqrt{(-2)^2 - 4*2*(-24)}] / (2*2)\).

Step 4 ：By solving the equation, we get two solutions: -3.0 and 4.0. However, since the length of a side of a triangle cannot be negative, we discard the negative solution.

Step 5 ：Therefore, the length of the longer leg of the triangle is 4 inches. Since the shortest leg is 1 inch shorter than the other leg, the length of the shortest leg is 3 inches.

Step 6 ：Final Answer: The shortest leg is \(\boxed{3}\) inches long. The other leg is \(\boxed{4}\) inches long.