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 complex numbers
Let:
- \( A = z \)
- \( B = \omega z \)
- \( C = z + \omega z \)

### Step 2: Understand the properties of \( \omega \)
The complex number \( \omega \) is a cube root of unity, which satisfies the equation:
\[ 1 + \omega + \omega^2 = 0 \]
From this, we can derive that:
\[ \omega^2 = -1 - \omega \]

### Step 3: Calculate the lengths of the sides
The lengths of the sides of the triangle can be calculated as follows:
- Length \( AB = |B - A| = |\omega z - z| = |z(\omega - 1)| \)
- Length \( BC = |C - B| = |(z + \omega z) - \omega z| = |z| \)
- Length \( CA = |A - C| = |z - (z + \omega z)| = |-\omega z| = |z| \)

### Step 4: Simplify the lengths
Since \( |z| \) is a common factor, we can express the lengths as:
- \( AB = |z| |\omega - 1| \)
- \( BC = |z| \)
- \( CA = |z| \)

### Step 5: Determine the equality of sides
To show that the triangle is equilateral, we need to show that all sides are equal. We know:
- \( BC = CA = |z| \)

Now, we need to check if \( AB = |z| \):
\[ |\omega - 1| = 1 \]
This holds true because \( \omega \) lies on the unit circle in the complex plane.

### Step 6: Area of the triangle
Since all sides are equal, the triangle is equilateral. The area \( A \) of an equilateral triangle with side length \( s \) is given by:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
In our case, the side length \( s = |z| \), so:
\[ A = \frac{\sqrt{3}}{4} |z|^2 \]

### Final Answer
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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