1. **State the problem:** We have a right triangle with the shorter leg as $(x - 9)$ inches, the longer leg as $x$ inches, and the hypotenuse as $(x + 9)$ inches. We need to find the lengths of the legs and the hypotenuse. 2. **Formula used:** For a right triangle, the Pythagorean theorem applies: $$ (\text{shorter leg})^2 + (\text{longer leg})^2 = (\text{hypotenuse})^2 $$ 3. **Set up the equation:** $$ (x - 9)^2 + x^2 = (x + 9)^2 $$ 4. **Expand each term:** $$ (x - 9)^2 = x^2 - 18x + 81 $$ $$ x^2 = x^2 $$ $$ (x + 9)^2 = x^2 + 18x + 81 $$ 5. **Substitute back into the equation:** $$ x^2 - 18x + 81 + x^2 = x^2 + 18x + 81 $$ 6. **Combine like terms:** $$ 2x^2 - 18x + 81 = x^2 + 18x + 81 $$ 7. **Bring all terms to one side:** $$ 2x^2 - 18x + 81 - x^2 - 18x - 81 = 0 $$ $$ x^2 - 36x = 0 $$ 8. **Factor the equation:** $$ x(x - 36) = 0 $$ 9. **Solve for $x$:** $$ x = 0 \quad \text{or} \quad x = 36 $$ Since length cannot be zero, $x = 36$ inches. 10. **Find the lengths:** - Longer leg: $x = 36$ inches - Shorter leg: $x - 9 = 36 - 9 = 27$ inches - Hypotenuse: $x + 9 = 36 + 9 = 45$ inches **Final answer:** - Shorter leg = 27 inches - Longer leg = 36 inches - Hypotenuse = 45 inches