1. **State the problem:** We have a right triangle with legs 8 and 32, and hypotenuse $x$. We need to find the value of $x$. 2. **Formula used:** In a right triangle, the Pythagorean theorem states: $$x^2 = a^2 + b^2$$ where $x$ is the hypotenuse and $a$, $b$ are the legs. 3. **Apply the formula:** $$x^2 = 8^2 + 32^2$$ $$x^2 = 64 + 1024$$ $$x^2 = 1088$$ 4. **Simplify the square root:** Factor 1088 to simplify: $$1088 = 64 \times 17$$ So, $$x = \sqrt{1088} = \sqrt{64 \times 17} = \sqrt{64} \times \sqrt{17} = 8\sqrt{17}$$ 5. **Check the options:** - A: $29\sqrt{2}$ - B: $31\sqrt{3}$ - C: $16\sqrt{5}$ - D: $13\sqrt{21}$ None of these match $8\sqrt{17}$ exactly, so let's verify if the hypotenuse is labeled $x$ or if $x$ is a leg. Since the problem states the hypotenuse is labeled $x$, the value is $8\sqrt{17}$. **Final answer:** $x = 8\sqrt{17}$ (not listed among the options).