1. **State the problem:** We have a right triangle $\triangle XYZ$ with right angle at $Y$. The sides $XY$ and $YZ$ are both 5 units. $YW$ is the altitude from $Y$ perpendicular to hypotenuse $XZ$. We need to find the length of $YW$. 2. **Recall the formula:** In a right triangle, the altitude to the hypotenuse can be found using the formula: $$YW = \frac{XY \times YZ}{XZ}$$ where $XZ$ is the hypotenuse. 3. **Find the hypotenuse $XZ$:** Using the Pythagorean theorem: $$XZ = \sqrt{XY^2 + YZ^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$$ 4. **Calculate $YW$:** Substitute values into the altitude formula: $$YW = \frac{5 \times 5}{5\sqrt{2}} = \frac{25}{5\sqrt{2}}$$ 5. **Simplify the fraction:** $$YW = \frac{\cancel{25}}{\cancel{5}\sqrt{2}} = \frac{5}{\sqrt{2}}$$ 6. **Rationalize the denominator:** $$YW = \frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2}$$ **Final answer:** $$YW = \frac{5\sqrt{2}}{2}$$