If a circle is inscribed in a square of side 10, so that the circle touches the four sides of the square internally then radius of the circle is

Options

A. 10
B. `5sqrt(2)`
C. `10sqrt(2)`
D. 5

Answer

To find the radius of a circle inscribed in a square with a side length of 10 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a square with a side length of 10 cm, and we need to find the radius of the inscribed circle that touches all four sides of the square.

2. **Draw the Diagram**: Visualize or draw a square and inscribe a circle inside it. The circle should touch all four sides of the square.

3. **Identify Key Relationships**: The radius of the circle (let's denote it as \( R \)) will be the distance from the center of the circle to any side of the square. Since the circle is inscribed, it will touch each side of the square at exactly one point.

4. **Use the Properties of the Square**: The distance from the center of the square to any side is equal to the radius of the inscribed circle. The center of the square divides the side into two equal segments.

5. **Calculate the Radius**:
- The total length of one side of the square is 10 cm.
- Since the circle touches both sides of the square, we can express this as:
\[ R + R = \text{side length of the square} \]
\[ 2R = 10 \text{ cm} \]
- Now, solve for \( R \):
\[ R = \frac{10}{2} = 5 \text{ cm} \]

6. **Conclusion**: The radius of the inscribed circle is 5 cm.

### Final Answer:
The radius of the circle is \( 5 \) cm.