A square is inscribed in a quarter circle in such a manner that two of its adjacent vertices lie on two radli and are equidistant from the centre. The other two vertices lie on the circular arc. If the length of the side of the square is 20 cm, find the radius of the circle. (in cm.)

Options

A. `10sqrt(10)`
B. `20sqrt(10)`
C. `10sqrt(20)`
D. `10sqrt(30)`

Answer

To solve the problem of finding the radius of the circle in which a square of side length 20 cm is inscribed in a quarter circle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have a quarter circle with a square inscribed in it. The square has two vertices on the radii of the quarter circle and the other two vertices on the arc of the quarter circle.

2. **Labeling the Square**:
- Let the side length of the square be \( s = 20 \) cm.
- Label the center of the quarter circle as \( O \), and the vertices of the square as \( A \), \( B \), \( C \), and \( D \) such that \( A \) and \( B \) are on the radii and \( C \) and \( D \) are on the arc.

3. **Finding the Diagonal of the Square**:
- The diagonal \( AC \) of the square can be calculated using the formula for the diagonal of a square:
\[ \text{Diagonal} = s\sqrt{2} = 20\sqrt{2} \text{ cm} \]

4. **Identifying the Radius**:
- The radius of the quarter circle \( r \) can be expressed in terms of the diagonal of the square. The diagonal \( AC \) extends from the center \( O \) to point \( C \) on the arc. Therefore, the radius \( r \) is equal to the length of the diagonal:
\[ r = 20\sqrt{2} \text{ cm} \]

5. **Calculating the Radius**:
- To find the numerical value of the radius, we calculate:
\[ r = 20\sqrt{2} \approx 20 \times 1.414 \approx 28.28 \text{ cm} \]

6. **Final Result**:
- The radius of the quarter circle is approximately \( 28.28 \) cm.

### Final Answer:
The radius of the circle is \( 20\sqrt{2} \) cm or approximately \( 28.28 \) cm.