Each side of a rhombus is 15 cm and the length of one its diagonals is 24 cm. The area of the rhombus is

Options

A. `432 cm^(2)`
B. `216 cm^(2)`
C. `180 cm^(2)`
D. `144 cm^(2)`

Answer

To find the area of the rhombus given that each side is 15 cm and one diagonal is 24 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the properties of the rhombus**:
- A rhombus has all sides equal, and its diagonals bisect each other at right angles.
- Let the diagonals be \(d_1\) and \(d_2\). We know \(d_1 = 24 \, \text{cm}\).

2. **Use the relationship between the sides and diagonals**:
- The formula for the area of a rhombus using the diagonals is:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
- We need to find the length of the second diagonal \(d_2\).

3. **Apply the Pythagorean theorem**:
- Since the diagonals bisect each other, we can form two right triangles. Each triangle will have legs of length \( \frac{d_1}{2} \) and \( \frac{d_2}{2} \), and the hypotenuse is the side of the rhombus (15 cm).
- Therefore, we can write:
\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = 15^2 \]
- Plugging in \(d_1 = 24\):
\[ \left(\frac{24}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = 15^2 \]
\[ 12^2 + \left(\frac{d_2}{2}\right)^2 = 15^2 \]
\[ 144 + \left(\frac{d_2}{2}\right)^2 = 225 \]

4. **Solve for \(d_2\)**:
- Rearranging the equation:
\[ \left(\frac{d_2}{2}\right)^2 = 225 - 144 \]
\[ \left(\frac{d_2}{2}\right)^2 = 81 \]
\[ \frac{d_2}{2} = 9 \quad \Rightarrow \quad d_2 = 18 \, \text{cm} \]

5. **Calculate the area of the rhombus**:
- Now that we have both diagonals, we can calculate the area:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 24 \times 18 \]
\[ \text{Area} = \frac{1}{2} \times 432 = 216 \, \text{cm}^2 \]

### Final Answer:
The area of the rhombus is \(216 \, \text{cm}^2\).