1. **Problem statement:** Given a right triangle with a hypotenuse of length $5\sqrt{2}$ and one angle of $45^\circ$, find the lengths of the legs $u$ and $v$. 2. **Formula and rules:** In a right triangle with a $45^\circ$ angle, the triangle is isosceles right-angled, meaning the legs $u$ and $v$ are equal. 3. The relationship between the legs and the hypotenuse in a $45^\circ-45^\circ-90^\circ$ triangle is: $$ \text{hypotenuse} = u \sqrt{2} $$ 4. Substitute the given hypotenuse length: $$ 5\sqrt{2} = u \sqrt{2} $$ 5. Divide both sides by $\sqrt{2}$: $$ \frac{5\cancel{\sqrt{2}}}{\cancel{\sqrt{2}}} = u $$ 6. Simplify: $$ u = 5 $$ 7. Since $u = v$ in this triangle, both legs are 5. **Final answer:** $u = 5$, $v = 5$ which corresponds to option D.