A rectangular painting has a diagonal measure of 13 inches and an area of 60 square inches. Use the formula for the area of a rectangle and the Pythagorean Theorem to find the length and width of the painting. The length (the longer side) is and the width (the shorter side) is (Simplify your answers.)

Step 1 ：We know that the area of a rectangle is given by the formula length * width = area. We also know that the diagonal of a rectangle forms a right triangle with the length and width, so we can use the Pythagorean theorem (\(a^2 + b^2 = c^2\)) to find the length and width. In this case, the diagonal (c) is 13 inches and the area is 60 square inches. We can set up the following system of equations to solve for the length and width:

Step 2 ：\[\begin{align*} \text{length} \times \text{width} &= 60 \\ \text{length}^2 + \text{width}^2 &= 13^2 \end{align*}\]

Step 3 ：We can solve this system of equations to find the length and width of the painting.

Step 4 ：The solution contains four pairs of values, two of which are negative. Since the length and width of a rectangle cannot be negative, we discard the negative solutions. The length is the larger value and the width is the smaller value.

Step 5 ：So, the length of the painting is 12 inches and the width is 5 inches.

Step 6 ：Final Answer: The length of the painting is \(\boxed{12}\) inches and the width is \(\boxed{5}\) inches.