A reactangle with one side 4 cm is inscribed in a circle of radius 2.5 cm. Find the area of the rectangle

Answer

To find the area of the rectangle inscribed in a circle with a radius of 2.5 cm, where one side of the rectangle is 4 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- One side of the rectangle (AD) = 4 cm
- Radius of the circle (R) = 2.5 cm

2. **Calculate the Diameter of the Circle:**
- The diameter (D) of the circle is given by the formula:
\[ D = 2 \times R \]
- Substituting the value of the radius:
\[ D = 2 \times 2.5 = 5 \text{ cm} \]

3. **Use the Pythagorean Theorem:**
- In a rectangle inscribed in a circle, the diagonal of the rectangle is equal to the diameter of the circle. Let the other side of the rectangle (AB) be denoted as \( x \).
- According to the Pythagorean theorem in triangle ABD:
\[ BD^2 = AB^2 + AD^2 \]
- Here, \( BD \) is the diameter of the circle, which is 5 cm, and \( AD \) is 4 cm.
- Therefore, we have:
\[ 5^2 = x^2 + 4^2 \]

4. **Substitute and Solve for \( x \):**
- Substitute the values:
\[ 25 = x^2 + 16 \]
- Rearranging gives:
\[ x^2 = 25 - 16 = 9 \]
- Taking the square root:
\[ x = \sqrt{9} = 3 \text{ cm} \]

5. **Calculate the Area of the Rectangle:**
- The area (A) of the rectangle is given by:
\[ A = AB \times AD \]
- Substituting the values:
\[ A = 3 \text{ cm} \times 4 \text{ cm} = 12 \text{ cm}^2 \]

### Final Answer:
The area of the rectangle is \( 12 \text{ cm}^2 \).

---