1. **State the problem:** We have a right triangle with angles 45°, 30°, and 90°, and sides labeled with $p$ opposite the 45° angle and $10\sqrt{3}$ adjacent to the 30° angle. We need to find the value of $p$ in simplest radical form. 2. **Identify the triangle type and relationships:** The triangle has angles 30°, 45°, and 90°, which is unusual since the angles in a triangle sum to 180°. However, since the triangle is right-angled, the third angle must be 90°, so the angles are 30°, 45°, and 105°, which is impossible for a right triangle. Likely, the triangle is a right triangle with angles 30°, 60°, and 90°, or 45°, 45°, and 90°. Given the labels, we assume the triangle is right-angled with angles 30°, 60°, and 90°. 3. **Recall side ratios for a 30°-60°-90° triangle:** The sides opposite these angles are in the ratio $1 : \sqrt{3} : 2$ respectively. 4. **Assign sides based on the given:** The side adjacent to the 30° angle is $10\sqrt{3}$, which corresponds to the side opposite 60°, so the side opposite 30° is $p$. 5. **Set up the ratio:** Since the side opposite 60° is $10\sqrt{3}$, and the ratio of sides opposite 30° to 60° is $1 : \sqrt{3}$, we have $$\frac{p}{10\sqrt{3}} = \frac{1}{\sqrt{3}}$$ 6. **Solve for $p$:** $$p = 10\sqrt{3} \times \frac{1}{\sqrt{3}}$$ 7. **Simplify:** $$p = 10 \cancel{\sqrt{3}} \times \frac{1}{\cancel{\sqrt{3}}} = 10$$ **Final answer:** $$p = 10$$