A rectangle in inscribed in a circle of radius 5 cm. if the breadth of the rectangle is 6 cm. then find the length of the rectangle.

Explanation

To find the length of the rectangle inscribed in a circle with a radius of 5 cm and a breadth of 6 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- The rectangle is inscribed in a circle, meaning all four corners of the rectangle touch the circle. The radius of the circle is given as 5 cm.

2. **Identify the Dimensions**:
- Let the breadth of the rectangle (AD) be 6 cm.
- Let the length of the rectangle (CD) be denoted as L.

3. **Use the Properties of the Circle**:
- The diagonal of the rectangle is equal to the diameter of the circle. Since the radius is 5 cm, the diameter (D) is:
\[ D = 2 \times \text{radius} = 2 \times 5 = 10 \text{ cm} \]

4. **Apply the Pythagorean Theorem**:
- In a rectangle, the relationship between the length (L), breadth (B), and the diagonal (D) can be expressed using the Pythagorean theorem:
\[ D^2 = L^2 + B^2 \]
- Substituting the known values:
\[ 10^2 = L^2 + 6^2 \]
- This simplifies to:
\[ 100 = L^2 + 36 \]

5. **Solve for the Length (L)**:
- Rearranging the equation gives:
\[ L^2 = 100 - 36 \]
\[ L^2 = 64 \]
- Taking the square root of both sides:
\[ L = \sqrt{64} = 8 \text{ cm} \]

6. **Conclusion**:
- The length of the rectangle is therefore 8 cm.

### Final Answer:
The length of the rectangle is **8 cm**.