In a rhombus, the length of a diagonal is 6√3 and length of other diagonal is equal to side of the rhombus. Then, find the area of rhombus.

Options

A. `18 sqrt3`
B. `20sqrt3`
C. `15 sqrt3`
D. `16sqrt3`

Answer

To find the area of the rhombus given the lengths of its diagonals, we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has the following properties:
- All four sides are equal in length.
- The diagonals bisect each other at right angles (90 degrees).

### Step 2: Define the diagonals
Let the length of one diagonal \( AC = 6\sqrt{3} \) and the other diagonal \( BD \) be equal to the side of the rhombus, which we will denote as \( 2A \).

### Step 3: Bisect the diagonals
Since the diagonals bisect each other, we can denote the midpoints of the diagonals as follows:
- \( O \) is the intersection point of the diagonals.
- Therefore, \( AO = OC = \frac{AC}{2} = \frac{6\sqrt{3}}{2} = 3\sqrt{3} \).
- Similarly, \( BO = OD = \frac{BD}{2} = A \).

### Step 4: Use the Pythagorean theorem
In triangle \( ODC \), we can apply the Pythagorean theorem:
\[ DC^2 = OD^2 + OC^2 \]
Here, \( DC \) is the side of the rhombus, which is \( 2A \), \( OD = 3\sqrt{3} \), and \( OC = A \). Thus, we have:
\[ (2A)^2 = (3\sqrt{3})^2 + A^2 \]
\[ 4A^2 = 27 + A^2 \]

### Step 5: Solve for \( A \)
Rearranging the equation gives:
\[ 4A^2 - A^2 = 27 \]
\[ 3A^2 = 27 \]
\[ A^2 = 9 \]
\[ A = 3 \]

### Step 6: Find the length of the other diagonal
Now that we have \( A \), we can find the length of the other diagonal \( AC \):
\[ AC = 2A = 2 \times 3 = 6 \]

### Step 7: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
Where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. Substituting the values we have:
\[ \text{Area} = \frac{1}{2} \times (6\sqrt{3}) \times 6 \]
\[ = \frac{1}{2} \times 36\sqrt{3} \]
\[ = 18\sqrt{3} \]

### Final Answer
The area of the rhombus is \( 18\sqrt{3} \) square units.