Show that the points representing the complex numbers `(3+ 3i), (-3- 3i) and (-3 sqrt3 + 3 sqrt3i)` on the Argand plane are the vertices of an equilateral triangle

Answer

To show that the points representing the complex numbers \( z_1 = 3 + 3i \), \( z_2 = -3 - 3i \), and \( z_3 = -3\sqrt{3} + 3\sqrt{3}i \) on the Argand plane are the vertices of an equilateral triangle, we need to calculate the distances between each pair of points and verify that they are equal.

### Step 1: Calculate the distance \( AB \) between \( z_1 \) and \( z_2 \)

The distance \( AB \) can be calculated using the modulus of the difference of the complex numbers:
\[ AB = |z_1 - z_2| = |(3 + 3i) - (-3 - 3i)| \]
Calculating this gives:
\[ AB = |3 + 3i + 3 + 3i| = |6 + 6i| \]
Now, we find the modulus:
\[ AB = \sqrt{(6)^2 + (6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} \]

### Step 2: Calculate the distance \( BC \) between \( z_2 \) and \( z_3 \)

Next, we calculate the distance \( BC \):
\[ BC = |z_2 - z_3| = |(-3 - 3i) - (-3\sqrt{3} + 3\sqrt{3}i)| \]
Calculating this gives:
\[ BC = |-3 + 3\sqrt{3} - 3i - 3\sqrt{3}i| = |(-3 + 3\sqrt{3}) + (-3 - 3\sqrt{3})i| \]
Now, we find the modulus:
\[ BC = \sqrt{(-3 + 3\sqrt{3})^2 + (-3 - 3\sqrt{3})^2} \]
Calculating each square:
\[ (-3 + 3\sqrt{3})^2 = 9 - 18\sqrt{3} + 27 = 36 - 18\sqrt{3} \]
\[ (-3 - 3\sqrt{3})^2 = 9 + 18\sqrt{3} + 27 = 36 + 18\sqrt{3} \]
Adding these:
\[ BC = \sqrt{(36 - 18\sqrt{3}) + (36 + 18\sqrt{3})} = \sqrt{72} = 6\sqrt{2} \]

### Step 3: Calculate the distance \( AC \) between \( z_1 \) and \( z_3 \)

Now, we calculate the distance \( AC \):
\[ AC = |z_1 - z_3| = |(3 + 3i) - (-3\sqrt{3} + 3\sqrt{3}i)| \]
Calculating this gives:
\[ AC = |3 + 3\sqrt{3} + (3 - 3\sqrt{3})i| = |(3 + 3\sqrt{3}) + (3 - 3\sqrt{3})i| \]
Now, we find the modulus:
\[ AC = \sqrt{(3 + 3\sqrt{3})^2 + (3 - 3\sqrt{3})^2} \]
Calculating each square:
\[ (3 + 3\sqrt{3})^2 = 9 + 18\sqrt{3} + 27 = 36 + 18\sqrt{3} \]
\[ (3 - 3\sqrt{3})^2 = 9 - 18\sqrt{3} + 27 = 36 - 18\sqrt{3} \]
Adding these:
\[ AC = \sqrt{(36 + 18\sqrt{3}) + (36 - 18\sqrt{3})} = \sqrt{72} = 6\sqrt{2} \]

### Conclusion

Since \( AB = BC = AC = 6\sqrt{2} \), all sides of the triangle are equal. Therefore, the points representing the complex numbers \( (3 + 3i) \), \( (-3 - 3i) \), and \( (-3\sqrt{3} + 3\sqrt{3}i) \) are the vertices of an equilateral triangle.