1. **Stating the problem:** We have a right triangle with a hypotenuse of length 4 and one angle of 45°. We want to find the length of the side opposite this 45° angle, labeled $x$. 2. **Formula and rules:** In a right triangle, the side lengths relate to angles via trigonometric functions. For angle $\theta$, the sine function is defined as: $$\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$$ 3. **Apply the formula:** Here, $\theta = 45^\circ$, the hypotenuse is 4, and the opposite side is $x$. So: $$\sin(45^\circ) = \frac{x}{4}$$ 4. **Evaluate $\sin(45^\circ)$:** We know that: $$\sin(45^\circ) = \frac{\sqrt{2}}{2}$$ 5. **Solve for $x$:** $$x = 4 \times \sin(45^\circ) = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2}$$ 6. **Conclusion:** The length of side $x$ is $2\sqrt{2}$.