1. **State the problem:** We have a right triangle with hypotenuse length 29 feet. One leg is 19 feet less than twice the other leg. We need to find the lengths of both legs. 2. **Set variables:** Let the shorter leg be $x$ feet. Then the other leg is $2x - 19$ feet. 3. **Use the Pythagorean theorem:** For a right triangle with legs $a$ and $b$ and hypotenuse $c$, the relation is: $$a^2 + b^2 = c^2$$ Here: $$x^2 + (2x - 19)^2 = 29^2$$ 4. **Expand and simplify:** $$x^2 + (2x - 19)^2 = 841$$ $$x^2 + (4x^2 - 76x + 361) = 841$$ $$5x^2 - 76x + 361 = 841$$ 5. **Bring all terms to one side:** $$5x^2 - 76x + 361 - 841 = 0$$ $$5x^2 - 76x - 480 = 0$$ 6. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=5$, $b=-76$, $c=-480$. Calculate discriminant: $$\Delta = (-76)^2 - 4 \times 5 \times (-480) = 5776 + 9600 = 15376$$ Calculate roots: $$x = \frac{76 \pm \sqrt{15376}}{10}$$ $$\sqrt{15376} = 124$$ So: $$x = \frac{76 \pm 124}{10}$$ Two solutions: - $$x = \frac{76 + 124}{10} = \frac{200}{10} = 20$$ - $$x = \frac{76 - 124}{10} = \frac{-48}{10} = -4.8$$ (not valid since length can't be negative) 7. **Find the other leg:** $$2x - 19 = 2 \times 20 - 19 = 40 - 19 = 21$$ **Final answer:** The legs are 20 feet and 21 feet.