The lengths of one side of a rhombus and one of two diagonals are 6 cm each. The area of the rhombus (in `cm^2` ) is:

Options

A. `27 sqrt3`
B. 18
C. `9 sqrt3`
D. `18 sqrt3`

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**:
- The length of one side of the rhombus (AB) is given as 6 cm.
- One diagonal (AC) is also given as 6 cm.

2. **Understand the Properties of the Rhombus**:
- In a rhombus, the diagonals bisect each other at right angles. Let's denote the diagonals as AC and BD.
- Let O be the intersection point of the diagonals. Thus, AO = OC = AC/2 = 6/2 = 3 cm.

3. **Apply the Pythagorean Theorem**:
- In triangle AOB, we can apply the Pythagorean theorem since AO and OB are perpendicular to each other.
- According to the theorem: \( AB^2 = AO^2 + OB^2 \)
- Substituting the known values: \( 6^2 = 3^2 + OB^2 \)
- This simplifies to: \( 36 = 9 + OB^2 \)
- Therefore, \( OB^2 = 36 - 9 = 27 \)
- Taking the square root gives us: \( OB = \sqrt{27} = 3\sqrt{3} \) cm.

4. **Calculate the Length of the Other Diagonal (BD)**:
- Since O is the midpoint of BD, we have: \( BD = 2 \times OB = 2 \times 3\sqrt{3} = 6\sqrt{3} \) cm.

5. **Use the Area Formula for a Rhombus**:
- 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.
- Here, \( d_1 = AC = 6 \) cm and \( d_2 = BD = 6\sqrt{3} \) cm.
- Substituting these values into the formula gives:
\[ A = \frac{1}{2} \times 6 \times 6\sqrt{3} = 18\sqrt{3} \text{ cm}^2. \]

6. **Final Result**:
- The area of the rhombus is \( 18\sqrt{3} \) cm².