The area of a rhombus is `120 cm^(2)` and one of its diagonals is 24 cm. Each side of the rhombus is

Options

A. 10 cm
B. 13 cm
C. 12 cm
D. 15 cm

Answer

To find the length of each side of the rhombus given the area and one diagonal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Area Formula**:
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

2. **Substitute the Known Values**:
We know the area \( A = 120 \, \text{cm}^2 \) and one diagonal \( d_1 = 24 \, \text{cm} \). We need to find the other diagonal \( d_2 \):
\[ 120 = \frac{1}{2} \times 24 \times d_2 \]

3. **Solve for the Unknown Diagonal**:
Multiply both sides by 2 to eliminate the fraction:
\[ 240 = 24 \times d_2 \]
Now, divide both sides by 24:
\[ d_2 = \frac{240}{24} = 10 \, \text{cm} \]

4. **Draw the Rhombus**:
Now that we have both diagonals, \( d_1 = 24 \, \text{cm} \) and \( d_2 = 10 \, \text{cm} \), we can visualize the rhombus. The diagonals bisect each other at right angles.

5. **Calculate the Length of Each Side**:
When the diagonals intersect, they form four right triangles. Each triangle has legs of lengths \( \frac{d_1}{2} \) and \( \frac{d_2}{2} \):
\[ \frac{d_1}{2} = \frac{24}{2} = 12 \, \text{cm} \]
\[ \frac{d_2}{2} = \frac{10}{2} = 5 \, \text{cm} \]

6. **Apply the Pythagorean Theorem**:
To find the length of each side \( s \) of the rhombus, we use the Pythagorean theorem:
\[ s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Substituting the values:
\[ s^2 = 12^2 + 5^2 = 144 + 25 = 169 \]
Therefore, taking the square root:
\[ s = \sqrt{169} = 13 \, \text{cm} \]

7. **Conclusion**:
Each side of the rhombus is \( 13 \, \text{cm} \).