If z is any complex number, then the area of the triangle formed by the complex number z, wz and z+wz as its sides, is

Options

A. `1/2|z|^(2)`
B. `3/2|z|^(2)`
C. `sqrt(3)/4|z|^(2)`
D. `1/2|z|^(2)`

Explanation

To find the area of the triangle formed by the complex numbers \( z \), \( \omega z \), and \( z + \omega z \), we can follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- Vertex A: \( z \)
- Vertex B: \( \omega z \)
- Vertex C: \( z + \omega z \)

### Step 2: Calculate the lengths of the sides
To find the lengths of the sides of the triangle, we need to calculate the distances between the vertices:
1. Length of side AB:
\[ |z - \omega z| = |z(1 - \omega)| \]
2. Length of side BC:
\[ |\omega z - (z + \omega z)| = |-\omega z| = |\omega||z| \]
3. Length of side CA:
\[ |(z + \omega z) - z| = |\omega z| = |\omega||z| \]

### Step 3: Simplify the lengths
From the calculations:
- Length AB: \( |z(1 - \omega)| \)
- Length BC: \( |\omega||z| \)
- Length CA: \( |\omega||z| \)

### Step 4: Determine if the triangle is equilateral
Since the lengths BC and CA are equal, we check if AB is also equal to these lengths:
- If \( |1 - \omega| = |\omega| \), then the triangle is equilateral.

### Step 5: Calculate the area of the triangle
The area \( A \) of an equilateral triangle with side length \( s \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
In our case, the side length \( s \) can be taken as \( |z| \) (assuming \( |1 - \omega| = |\omega| \)):
\[ A = \frac{\sqrt{3}}{4} |z|^2 \]

### Conclusion
Thus, the area of the triangle formed by the complex numbers \( z \), \( \omega z \), and \( z + \omega z \) is:
\[ \frac{\sqrt{3}}{4} |z|^2 \]

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