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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**.

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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!

A square is inscribed in a circle. The radius of the circle is (20√2)/2 inches. What is the side length, in inches, of the square? A) 20 B) (20√2)/2 C) 20√2 D) 40

Find the side length of a square inscribed in a circle with a radius of (20√2)/2 inches. This geometry problem explores the relationship between the circle's radius and the side length of the square using the Pythagorean theorem. Suitable for students in Grades 9-10.

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