If `|z_1 |=|z_2|=|z_3|` = 1 and `z_1 +z_2+z_3` =0 then the area of the triangle whose vertices are `z_1 ,z_2 ,z_3`is

Options

A. `(3sqrt3)/4`
B. `sqrt3`
C. 1
D. none of these

Answer

To find the area of the triangle whose vertices are the complex numbers \( z_1, z_2, z_3 \) given that \( |z_1| = |z_2| = |z_3| = 1 \) and \( z_1 + z_2 + z_3 = 0 \), we can follow these steps:

### Step 1: Understand the conditions
We know that \( |z_1| = |z_2| = |z_3| = 1 \). This means that \( z_1, z_2, z_3 \) lie on the unit circle in the complex plane.

### Step 2: Find the centroid
The centroid \( G \) of the triangle formed by the points \( z_1, z_2, z_3 \) is given by:
\[ G = \frac{z_1 + z_2 + z_3}{3} \]
Since \( z_1 + z_2 + z_3 = 0 \), we have:
\[ G = \frac{0}{3} = 0 \]
This means that the centroid of the triangle is at the origin.

### Step 3: Identify the circumcenter
Since \( |z_1| = |z_2| = |z_3| = 1 \), the circumcenter of the triangle is also at the origin (the center of the unit circle).

### Step 4: Determine the type of triangle
When the centroid and circumcenter coincide, it indicates that the triangle is equilateral. Thus, \( z_1, z_2, z_3 \) form an equilateral triangle.

### Step 5: Calculate the side length
Let the side length of the equilateral triangle be \( a \). The distance between any two points \( z_i \) and \( z_j \) on the unit circle can be calculated using the formula:
\[ |z_i - z_j| = 2 \sin\left(\frac{\theta}{2}\right) \]
where \( \theta \) is the angle subtended at the center of the circle by the points \( z_i \) and \( z_j \). For an equilateral triangle, \( \theta = 120^\circ \):
\[ |z_1 - z_2| = 2 \sin\left(\frac{120^\circ}{2}\right) = 2 \sin(60^\circ) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]
Thus, the side length \( a = \sqrt{3} \).

### Step 6: Calculate the area of the triangle
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \( a = \sqrt{3} \):
\[ A = \frac{\sqrt{3}}{4} (\sqrt{3})^2 = \frac{\sqrt{3}}{4} \cdot 3 = \frac{3\sqrt{3}}{4} \]

### Final Answer
The area of the triangle whose vertices are \( z_1, z_2, z_3 \) is:
\[ \frac{3\sqrt{3}}{4} \]