If the perimeter of a rhombus is 80 cm. and one of its diagonals is 24 cm, then what is the area (in `cm^(2)`) of the rhombus ?

Options

A. 218
B. 192
C. 384
D. 768

Answer

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

### Step-by-Step Solution:

1. **Understand the properties of a rhombus**: A rhombus has all sides equal, and its diagonals bisect each other at right angles.

2. **Calculate the length of one side**:
- Given the perimeter of the rhombus is 80 cm, we can find the length of one side (let's denote it as \( a \)).
- The formula for the perimeter \( P \) of a rhombus is:
\[ P = 4a \]
- Setting up the equation:
\[ 4a = 80 \]
- Solving for \( a \):
\[ a = \frac{80}{4} = 20 \text{ cm} \]

3. **Identify the diagonals**:
- Let \( d_1 \) be the length of the given diagonal, which is 24 cm.
- Let \( d_2 \) be the length of the other diagonal, which we need to find.

4. **Use the properties of the diagonals**:
- The diagonals bisect each other at right angles. Thus, if we denote the half-lengths of the diagonals as \( \frac{d_1}{2} \) and \( \frac{d_2}{2} \):
\[ \frac{d_1}{2} = \frac{24}{2} = 12 \text{ cm} \]
- Let \( \frac{d_2}{2} = x \).

5. **Apply the Pythagorean theorem**:
- In the right triangle formed by half of each diagonal and the side of the rhombus:
\[ a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
- Substituting the known values:
\[ 20^2 = 12^2 + x^2 \]
\[ 400 = 144 + x^2 \]
- Rearranging gives:
\[ x^2 = 400 - 144 = 256 \]
- Taking the square root:
\[ x = \sqrt{256} = 16 \text{ cm} \]
- Therefore, the length of the second diagonal \( d_2 \) is:
\[ d_2 = 2x = 2 \times 16 = 32 \text{ cm} \]

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:
\[ A = \frac{1}{2} \times 24 \times 32 \]
\[ A = \frac{1}{2} \times 768 = 384 \text{ cm}^2 \]

### Final Answer:
The area of the rhombus is \( 384 \text{ cm}^2 \).