1. **Problem statement:** We have a right triangle with a hypotenuse of length 10, an angle of 30°, and the side adjacent to the 30° angle labeled as $5\sqrt{3}$. We need to find the length of the side opposite the 30° angle, denoted as $x$. 2. **Relevant formula:** In a right triangle, the sides relate to the angles as follows: - Opposite side = Hypotenuse $\times \sin(\theta)$ - Adjacent side = Hypotenuse $\times \cos(\theta)$ 3. **Given:** Hypotenuse $= 10$, angle $\theta = 30^\circ$, adjacent side $= 5\sqrt{3}$. 4. **Check the adjacent side:** Using $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, $$\text{Adjacent} = 10 \times \cos(30^\circ) = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}$$ This matches the given adjacent side, confirming the setup. 5. **Find the opposite side $x$:** Using $\sin(30^\circ) = \frac{1}{2}$, $$x = 10 \times \sin(30^\circ) = 10 \times \frac{1}{2} = 5$$ 6. **Answer:** The length of the side opposite the 30° angle is $\boxed{5}$.