The perimeter of a non-square rhombus is 20 cm. One of its diagonal is 8 cm. The area of the rhombus is

Options

A. 28 sq cm
B. 20 sq cm
C. 22 sq cm
D. 24 sq cm

Answer

To find the area of the rhombus given its perimeter and one diagonal, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Information:**
- Perimeter of the rhombus = 20 cm
- One diagonal (let's call it \( d_1 \)) = 8 cm

2. **Calculate the Length of One Side of the Rhombus:**
- The perimeter of a rhombus is given by the formula:
\[ \text{Perimeter} = 4 \times \text{side} \]
- Therefore, we can find the length of one side:
\[ \text{side} = \frac{\text{Perimeter}}{4} = \frac{20 \text{ cm}}{4} = 5 \text{ cm} \]

3. **Use the Properties of the Rhombus:**
- The diagonals of a rhombus bisect each other at right angles. Let the other diagonal be \( d_2 \).
- Since \( d_1 = 8 \text{ cm} \), half of \( d_1 \) is:
\[ \frac{d_1}{2} = \frac{8 \text{ cm}}{2} = 4 \text{ cm} \]

4. **Apply the Pythagorean Theorem:**
- In the right triangle formed by half of the diagonals and the side of the rhombus, we can apply the Pythagorean theorem:
\[ \text{side}^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
- Substituting the known values:
\[ 5^2 = 4^2 + \left(\frac{d_2}{2}\right)^2 \]
\[ 25 = 16 + \left(\frac{d_2}{2}\right)^2 \]
- Rearranging gives:
\[ \left(\frac{d_2}{2}\right)^2 = 25 - 16 = 9 \]
- Taking the square root:
\[ \frac{d_2}{2} = 3 \implies d_2 = 6 \text{ cm} \]

5. **Calculate the Area of the Rhombus:**
- The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
- Substituting the values of the diagonals:
\[ A = \frac{1}{2} \times 8 \text{ cm} \times 6 \text{ cm} = \frac{1}{2} \times 48 \text{ cm}^2 = 24 \text{ cm}^2 \]

### Final Answer:
The area of the rhombus is **24 cm²**.