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The problem you're asking involves a square inscribed in a semicircle. Let's break it down step by step.

### Problem restatement:

We have a semicircle with:
- Diameter \( AB \), and the center is at \( O \).
- The radius of the semicircle is \( R \).
- A square is inscribed within the semicircle.
- We need to find the area of the square in terms of \( R \), choosing from the provided options.

### Solution Approach:

1. **Geometry Setup:**
   - The diameter of the semicircle is \( 2R \).
   - The side of the square, let it be \( s \), will touch the flat side of the semicircle (the diameter) and two points on the arc of the semicircle.

2. **Key Geometrical Insight:**
   - The square is inscribed, meaning the top two corners of the square lie on the semicircle. By symmetry, the center of the square is directly below the center of the semicircle.
   - If we consider the full circle, a square inscribed inside a circle of radius \( R \) has its side length \( s = R\sqrt{2} \). However, since this is a semicircle, we need to adapt the geometry slightly.

3. **Relationship Between the Radius and the Square's Side:**
   Using some geometric and trigonometric reasoning (or formulas derived from such configurations), the side length of the square inscribed in a semicircle is \( s = \frac{2R}{\sqrt{5}} \).

4. **Finding the Area of the Square:**
   The area \( A \) of the square is given by \( A = s^2 \). So:

   \[
   A = \left(\frac{2R}{\sqrt{5}}\right)^2 = \frac{4R^2}{5}
   \]

Thus, the area of the inscribed square is \( \frac{4R^2}{5} \).

### Answer:
The correct option is **C)** \( \frac{4R^2}{5} \).

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Here are five related questions to explore this topic further:

1. How can we derive the side length of a square inscribed in a full circle?
2. What is the area of a circle, and how does it compare to the area of the inscribed square?
3. How does the relationship between the square and the semicircle change if the radius \( R \) increases or decreases?
4. Could we inscribe a rectangle in a semicircle? How would its area compare to that of the square?
5. What are the practical applications of inscribed shapes in real-world problems, such as design or architecture?

**Tip:** Understanding how geometric shapes fit within each other, like squares in circles or semicircles, helps you solve problems involving optimization and design.

En una semicircunferencia de diámetro AB y centro O, se inscribe un cuadrado. Si el radio de la semicircunferencia mide R, entonces el área de la región cuadrada inscrita es...

Find the area of a square inscribed in a semicircle with radius R. Learn the geometric properties and key formulas involved, such as the side length of an inscribed square and how it relates to the radius of the semicircle. Suitable for students in Grades 9-12.

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