One side of a rhombus is 13 cm and one of its diagonals is 24 cm. What is the area of the rhombus?

Options

A. `312 cm^2`
B. `156cm^2`
C. `120cm^2 `
D. `130cm^2`

Answer

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

### Step-by-Step Solution:

1. **Identify the given values**:
- Side of the rhombus (s) = 13 cm
- One diagonal (d1) = 24 cm

2. **Understand the properties of the rhombus**:
- The diagonals of a rhombus bisect each other at right angles.
- Let the second diagonal be d2.

3. **Divide the rhombus into right triangles**:
- When we draw the diagonals, they intersect at a point (let's call it O), forming four right triangles.
- Each diagonal is divided into two equal halves. Therefore, half of diagonal d1 (d1/2) = 24 cm / 2 = 12 cm.

4. **Use the Pythagorean theorem**:
- In one of the right triangles formed, we can apply the Pythagorean theorem:
\[ s^2 = \left(\frac{d1}{2}\right)^2 + \left(\frac{d2}{2}\right)^2 \]
- Plugging in the known values:
\[ 13^2 = 12^2 + \left(\frac{d2}{2}\right)^2 \]
\[ 169 = 144 + \left(\frac{d2}{2}\right)^2 \]
\[ \left(\frac{d2}{2}\right)^2 = 169 - 144 = 25 \]
\[ \frac{d2}{2} = 5 \]
- Therefore, the full length of diagonal d2 is:
\[ d2 = 2 \times 5 = 10 \text{ cm} \]

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 we have:
\[ A = \frac{1}{2} \times 24 \times 10 \]
\[ A = \frac{240}{2} = 120 \text{ cm}^2 \]

### Final Answer:
The area of the rhombus is **120 cm²**.