1. **State the problem:** Find the length of a leg $x$ in a right triangle with a $45^\circ$ angle and hypotenuse length 5. 2. **Recall the properties of a $45^\circ-45^\circ-90^\circ$ triangle:** The legs are congruent, and the hypotenuse is $\sqrt{2}$ times the length of each leg. 3. **Formula:** If the leg length is $x$, then hypotenuse $h = x\sqrt{2}$. 4. **Set up the equation:** Given $h=5$, we have $$5 = x\sqrt{2}$$ 5. **Solve for $x$:** $$x = \frac{5}{\sqrt{2}}$$ 6. **Rationalize the denominator:** $$x = \frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2}$$ 7. **Final answer:** The length of the leg is $\boxed{\frac{5\sqrt{2}}{2}}$. --- **Second problem:** Find the length of a leg of a $45^\circ-45^\circ-90^\circ$ triangle with hypotenuse 12 inches. 1. **Recall the formula:** Hypotenuse $h = x\sqrt{2}$ where $x$ is the leg length. 2. **Set up the equation:** $$12 = x\sqrt{2}$$ 3. **Solve for $x$:** $$x = \frac{12}{\sqrt{2}}$$ 4. **Rationalize the denominator:** $$x = \frac{12}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2}$$ 5. **Final answer:** The leg length is $\boxed{6\sqrt{2}}$ inches.