1. **State the problem:** We have a right triangle with three sides: the shorter leg, the longer leg, and the hypotenuse. - The shorter leg is 9 inches shorter than the longer leg. - The hypotenuse is 9 inches longer than the longer leg. We need to find the lengths of all three sides. 2. **Define variables:** Let the length of the longer leg be $x$ inches. Then: - Shorter leg = $x - 9$ - Hypotenuse = $x + 9$ 3. **Use the Pythagorean theorem:** For a right triangle, the sum of the squares of the legs equals the square of the hypotenuse: $$ (\text{shorter leg})^2 + (\text{longer leg})^2 = (\text{hypotenuse})^2 $$ Substitute the expressions: $$ (x - 9)^2 + x^2 = (x + 9)^2 $$ 4. **Expand each term:** $$ (x - 9)^2 = x^2 - 18x + 81 $$ $$ x^2 = x^2 $$ $$ (x + 9)^2 = x^2 + 18x + 81 $$ So the equation becomes: $$ x^2 - 18x + 81 + x^2 = x^2 + 18x + 81 $$ 5. **Combine like terms:** $$ 2x^2 - 18x + 81 = x^2 + 18x + 81 $$ 6. **Bring all terms to one side:** $$ 2x^2 - 18x + 81 - x^2 - 18x - 81 = 0 $$ Simplify: $$ x^2 - 36x = 0 $$ 7. **Factor the equation:** $$ x(x - 36) = 0 $$ 8. **Solve for $x$:** $$ x = 0 \quad \text{or} \quad x = 36 $$ Since side lengths must be positive, $x = 36$ inches. 9. **Find the other sides:** - Shorter leg = $36 - 9 = 27$ inches - Hypotenuse = $36 + 9 = 45$ inches 10. **Check with Pythagorean theorem:** $$ 27^2 + 36^2 = 729 + 1296 = 2025 $$ $$ 45^2 = 2025 $$ The equality holds, so the solution is correct. **Final answer:** - Length of the shorter leg: 27 inches - Length of the longer leg: 36 inches - Length of the hypotenuse: 45 inches