Find the area and nature of the triangle formed by the points represented by the complex numbers ` (3+3i),(-3-3i) and (-3sqrt(3)+3sqrt(3)i)`.

Explanation

To find the area and nature of the triangle formed by the points represented by the complex numbers \( z_1 = 3 + 3i \), \( z_2 = -3 - 3i \), and \( z_3 = -3\sqrt{3} + 3\sqrt{3}i \), we can follow these steps:

### Step 1: Identify the coordinates of the points
The complex numbers can be represented as points in the Cartesian plane:
- \( z_1 = 3 + 3i \) corresponds to the point \( (3, 3) \)
- \( z_2 = -3 - 3i \) corresponds to the point \( (-3, -3) \)
- \( z_3 = -3\sqrt{3} + 3\sqrt{3}i \) corresponds to the point \( (-3\sqrt{3}, 3\sqrt{3}) \)

### Step 2: Use the formula for the area of a triangle
The area \( A \) of a triangle formed by the points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates:
- \( x_1 = 3, y_1 = 3 \)
- \( x_2 = -3, y_2 = -3 \)
- \( x_3 = -3\sqrt{3}, y_3 = 3\sqrt{3} \)

The area becomes:
\[ A = \frac{1}{2} \left| 3(-3 - 3\sqrt{3}) + (-3)(3\sqrt{3} - 3) + (-3\sqrt{3})(3 - (-3)) \right| \]

### Step 3: Calculate the area
Calculating each term:
1. \( 3(-3 - 3\sqrt{3}) = -9 - 9\sqrt{3} \)
2. \( -3(3\sqrt{3} - 3) = -9\sqrt{3} + 9 \)
3. \( -3\sqrt{3}(3 + 3) = -18\sqrt{3} \)

Now, combine these:
\[ A = \frac{1}{2} \left| (-9 - 9\sqrt{3}) + (-9\sqrt{3} + 9) + (-18\sqrt{3}) \right| \]
\[ = \frac{1}{2} \left| -9 - 9\sqrt{3} - 9\sqrt{3} - 18\sqrt{3} + 9 \right| \]
\[ = \frac{1}{2} \left| -36\sqrt{3} \right| = 18\sqrt{3} \]

### Step 4: Determine the nature of the triangle
To determine the nature of the triangle, we calculate the lengths of the sides using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

1. **Distance \( z_1 \) to \( z_2 \)**:
\[ d_{12} = \sqrt{((-3) - 3)^2 + ((-3) - 3)^2} = \sqrt{(-6)^2 + (-6)^2} = \sqrt{36 + 36} = 6\sqrt{2} \]

2. **Distance \( z_2 \) to \( z_3 \)**:
\[ d_{23} = \sqrt{((-3\sqrt{3}) - (-3))^2 + (3\sqrt{3} - (-3))^2} \]
Calculating:
\[ = \sqrt{(-3\sqrt{3} + 3)^2 + (3\sqrt{3} + 3)^2} \]
Let \( a = 3 \) and \( b = 3\sqrt{3} \):
\[ = \sqrt{(3 - 3\sqrt{3})^2 + (3 + 3\sqrt{3})^2} = \sqrt{(3(1 - \sqrt{3}))^2 + (3(1 + \sqrt{3}))^2} \]
\[ = 6\sqrt{2} \]

3. **Distance \( z_3 \) to \( z_1 \)**:
\[ d_{31} = \sqrt{(3 - (-3\sqrt{3}))^2 + (3 - 3\sqrt{3})^2} \]
Calculating similarly yields:
\[ = 6\sqrt{2} \]

### Conclusion
Since all three sides are equal, the triangle is equilateral.

### Final Answer
- **Area of the triangle**: \( 18\sqrt{3} \) square units
- **Nature of the triangle**: Equilateral