1. **Problem Statement:** Find the value of $x$ and the lengths of the sides of a right triangle where the two shorter sides are $x$ cm and $(x - 7)$ cm, and the hypotenuse is $(x + 1)$ cm. 2. **Formula Used:** By the Pythagorean theorem for right triangles: $$\text{(leg)}^2 + \text{(leg)}^2 = \text{(hypotenuse)}^2$$ 3. **Set up the equation:** $$x^2 + (x - 7)^2 = (x + 1)^2$$ 4. **Expand each term:** $$x^2 + (x^2 - 14x + 49) = x^2 + 2x + 1$$ 5. **Combine like terms on the left:** $$x^2 + x^2 - 14x + 49 = x^2 + 2x + 1$$ 6. **Simplify:** $$2x^2 - 14x + 49 = x^2 + 2x + 1$$ 7. **Bring all terms to one side:** $$2x^2 - 14x + 49 - x^2 - 2x - 1 = 0$$ 8. **Simplify:** $$x^2 - 16x + 48 = 0$$ 9. **Factor the quadratic:** $$x^2 - 16x + 48 = (x - 12)(x - 4) = 0$$ 10. **Solve for $x$:** $$x - 12 = 0 \Rightarrow x = 12$$ $$x - 4 = 0 \Rightarrow x = 4$$ 11. **Check for valid side lengths:** - For $x = 12$: - shorter sides: $12$ cm and $12 - 7 = 5$ cm - hypotenuse: $12 + 1 = 13$ cm - Check Pythagorean theorem: $12^2 + 5^2 = 144 + 25 = 169$, and $13^2 = 169$ ✔ - For $x = 4$: - shorter sides: $4$ cm and $4 - 7 = -3$ cm (not possible, side length cannot be negative) 12. **Final answer:** The value of $x$ is $12$. The sides of the triangle are $12$ cm, $5$ cm, and $13$ cm.