The area of a rhombus whose one side and one diagonal measure 20 cm and 24 cm respectively, is

Options

A. `364 cm^(2)`
B. `374 cm^(2)`
C. `384 cm^(2)`
D. `394 cm^(2)`

Answer

To find the area of a rhombus given one side and one diagonal, we can use the following steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Side of the rhombus (s) = 20 cm
- One diagonal (d1) = 24 cm

2. **Use the Property of Rhombus**:
- In a rhombus, the diagonals bisect each other at right angles. Let the other diagonal be d2. The diagonals divide the rhombus into four right-angled triangles.

3. **Apply the Pythagorean Theorem**:
- In one of the right triangles formed by the diagonals, we can use the Pythagorean theorem:
\[ s^2 = \left(\frac{d1}{2}\right)^2 + \left(\frac{d2}{2}\right)^2 \]
- Substituting the known values:
\[ 20^2 = \left(\frac{24}{2}\right)^2 + \left(\frac{d2}{2}\right)^2 \]
\[ 400 = 12^2 + \left(\frac{d2}{2}\right)^2 \]
\[ 400 = 144 + \left(\frac{d2}{2}\right)^2 \]

4. **Solve for d2**:
- Rearranging the equation:
\[ \left(\frac{d2}{2}\right)^2 = 400 - 144 \]
\[ \left(\frac{d2}{2}\right)^2 = 256 \]
\[ \frac{d2}{2} = 16 \]
\[ d2 = 32 \text{ cm} \]

5. **Calculate the Area of the Rhombus**:
- The area (A) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d1 \times d2 \]
- Substituting the values of the diagonals:
\[ A = \frac{1}{2} \times 24 \times 32 \]
\[ A = \frac{1}{2} \times 768 \]
\[ A = 384 \text{ cm}^2 \]

### Final Answer:
The area of the rhombus is **384 cm²**.

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