1. **Problem Statement:** We have a special right triangle with angles 45°, 45°, and 90°, where the hypotenuse is given as $7\sqrt{2}$. We need to find the lengths of legs $x$ and $y$. 2. **Formula and Rules:** In a 45°-45°-90° triangle, the legs are congruent, and the hypotenuse is $\sqrt{2}$ times the length of each leg. This means: $$\text{hypotenuse} = x\sqrt{2} = y\sqrt{2}$$ where $x = y$. 3. **Find the legs:** Given the hypotenuse $7\sqrt{2}$, set up the equation: $$7\sqrt{2} = x\sqrt{2}$$ Divide both sides by $\sqrt{2}$: $$x = 7$$ Since $x = y$, we also have: $$y = 7$$ 4. **Check the options:** - $x = 7$, $y = 7\sqrt{2}$ (incorrect, $y$ should equal $x$) - $x = 14$, $y = 7\sqrt{2}$ (incorrect) - $x = 14\sqrt{2}$, $y = 7\sqrt{2}$ (incorrect) - $x = 7$, $y = 7$ (correct) **Final answer:** $x = 7$, $y = 7$