The problem states that a square is inscribed in a circle, and the radius of the circle is given as $\frac{20\sqrt{2}}{2}$ inches. We are asked to find the side length of the square.

Step-by-step solution:

1. Relationship between the square and the circle: When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. Let's call the side length of the square $s$. The diagonal of the square can be related to its side by the Pythagorean theorem: $\text{Diagonal of the square} = s\sqrt{2}$

2. Diameter of the circle: The diameter of the circle is twice the radius. Since the radius is given as $\frac{20\sqrt{2}}{2}$, the diameter is: $\text{Diameter of the circle} = 2 \times \frac{20\sqrt{2}}{2} = 20\sqrt{2}$

3. Equating the diagonal of the square to the diameter of the circle: Since the diagonal of the square is equal to the diameter of the circle: $s\sqrt{2} = 20\sqrt{2}$ Now, divide both sides by $\sqrt{2}$: $s = 20$

Thus, the side length of the square is $\boxed{20}$ inches, which corresponds to option A.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How would the side length of the square change if the radius of the circle were doubled?
2. Can you derive a general formula for the side length of a square inscribed in a circle with radius $r$?
3. What would be the area of the square with side length 20 inches?
4. How does the diagonal of a square relate to its side in terms of other geometrical shapes?
5. How does the concept of an inscribed square apply to other polygons?

Tip: For any square, the diagonal is always $\sqrt{2}$ times the side length. This is a useful property in problems involving inscribed squares!