1. **State the problem:** We have a right triangle with hypotenuse length 5 ft. The longer leg is 1 ft longer than the shorter leg. We need to find the lengths of the shorter leg and the longer leg. 2. **Set variables:** Let the shorter leg be $x$ ft. Then the longer leg is $x + 1$ ft. 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: $$x^2 + (x+1)^2 = 5^2$$ 4. **Expand and simplify:** $$x^2 + (x+1)^2 = 25$$ $$x^2 + (x^2 + 2x + 1) = 25$$ $$2x^2 + 2x + 1 = 25$$ 5. **Bring all terms to one side:** $$2x^2 + 2x + 1 - 25 = 0$$ $$2x^2 + 2x - 24 = 0$$ 6. **Divide entire equation by 2 to simplify:** $$\cancel{2}x^2 + \cancel{2}x - \cancel{24} = 0$$ $$x^2 + x - 12 = 0$$ 7. **Factor the quadratic:** $$x^2 + x - 12 = (x + 4)(x - 3) = 0$$ 8. **Solve for $x$:** $$x + 4 = 0 \Rightarrow x = -4$$ (not valid since length cannot be negative) $$x - 3 = 0 \Rightarrow x = 3$$ 9. **Find the longer leg:** $$x + 1 = 3 + 1 = 4$$ 10. **Check with Pythagorean theorem:** $$3^2 + 4^2 = 9 + 16 = 25 = 5^2$$ Correct. **Final answer:** - Length of the shorter leg: 3 ft - Length of the longer leg: 4 ft - Length of the hypotenuse: 5 ft