The perimeter of a rhombus is 20 cm. One of its diagonals measures 8 cm. The area of rhombus is

Options

A. `12 cm^(2)`
B. `24 cm^(2)`
C. `80 cm^(2)`
D. `40 cm^(2)`

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 denote it as \(d_1\)) = 8 cm

2. **Calculate the length of one side of the rhombus**:
\[ \text{Perimeter} = 4 \times \text{side} \]
Let the side of the rhombus be \(a\). Therefore,
\[ 4a = 20 \implies a = \frac{20}{4} = 5 \text{ cm} \]

3. **Use the properties of the rhombus**:
In a rhombus, the diagonals bisect each other at right angles. Let the other diagonal be \(d_2\). The diagonals divide the rhombus into four right-angled triangles.

4. **Calculate the lengths of the halves of the diagonals**:
Since \(d_1 = 8\) cm, each half of \(d_1\) is:
\[ \frac{d_1}{2} = \frac{8}{2} = 4 \text{ cm} \]
Let \(d_2\) be the other diagonal. Each half of \(d_2\) will be \(\frac{d_2}{2}\).

5. **Apply the Pythagorean theorem**:
In one of the right triangles formed by the diagonals, we have:
\[ a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Substituting the known values:
\[ 5^2 = 4^2 + \left(\frac{d_2}{2}\right)^2 \]
\[ 25 = 16 + \left(\frac{d_2}{2}\right)^2 \]
\[ \left(\frac{d_2}{2}\right)^2 = 25 - 16 = 9 \]
\[ \frac{d_2}{2} = 3 \implies d_2 = 6 \text{ cm} \]

6. **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:
\[ A = \frac{1}{2} \times 8 \times 6 = \frac{48}{2} = 24 \text{ cm}^2 \]

### Final Answer:
The area of the rhombus is \(24 \text{ cm}^2\).