1. **State the problem:** We have a right triangle with a vertical leg of length 10 inches and a horizontal leg of length 8 inches. We want to analyze or find properties related to this triangle. 2. **Formula used:** For a right triangle, the Pythagorean theorem applies: $$c^2 = a^2 + b^2$$ where $c$ is the hypotenuse, and $a$, $b$ are the legs. 3. **Apply the Pythagorean theorem:** $$c^2 = 10^2 + 8^2 = 100 + 64 = 164$$ 4. **Find the hypotenuse:** $$c = \sqrt{164} = \sqrt{4 \times 41} = 2\sqrt{41} \approx 12.81$$ inches. 5. **Explanation:** The hypotenuse is the longest side opposite the right angle. Using the Pythagorean theorem, we square the legs, sum them, and take the square root to find the hypotenuse length. **Final answer:** The hypotenuse length is approximately $12.81$ inches.