If four complex number `z,bar(z),bar(z)-2Re(bar(z))` and `z-2Re(z)` represent the vertices of a square of side 4-units in the Argand plane than find `|z|`.

Answer

To solve the problem, we need to find the modulus of the complex number \( z \) given that the four complex numbers \( z, \bar{z}, \bar{z} - 2\text{Re}(\bar{z}), z - 2\text{Re}(z) \) represent the vertices of a square with side length 4 units in the Argand plane.

### Step 1: Define the complex number \( z \)
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of \( z \).

**Hint:** Remember that the complex number can be expressed in terms of its real and imaginary parts.

### Step 2: Find \( \bar{z} \)
The conjugate of \( z \) is given by:
\[ \bar{z} = x - iy \]

**Hint:** The conjugate of a complex number is obtained by changing the sign of the imaginary part.

### Step 3: Calculate \( \bar{z} - 2\text{Re}(\bar{z}) \)
The real part of \( \bar{z} \) is \( x \). Thus, we have:
\[ \bar{z} - 2\text{Re}(\bar{z}) = (x - iy) - 2x = -x - iy \]

**Hint:** When subtracting \( 2\text{Re}(\bar{z}) \), you are effectively moving the point left by \( 2x \) on the real axis.

### Step 4: Calculate \( z - 2\text{Re}(z) \)
The real part of \( z \) is also \( x \). Therefore:
\[ z - 2\text{Re}(z) = (x + iy) - 2x = -x + iy \]

**Hint:** Similar to the previous step, this moves the point left by \( 2x \) on the real axis.

### Step 5: Identify the vertices of the square
The vertices of the square are:
1. \( z = x + iy \)
2. \( \bar{z} = x - iy \)
3. \( \bar{z} - 2\text{Re}(\bar{z}) = -x - iy \)
4. \( z - 2\text{Re}(z) = -x + iy \)

**Hint:** Visualizing these points on the Argand plane can help you understand their arrangement.

### Step 6: Calculate the distances between the vertices
The distance between \( z \) and \( \bar{z} \) is:
\[ \text{Distance} = |z - \bar{z}| = |(x + iy) - (x - iy)| = |2iy| = 2|y| \]
The distance between \( z \) and \( \bar{z} - 2\text{Re}(\bar{z}) \) is:
\[ \text{Distance} = |z - (-x - iy)| = |(x + iy) - (-x - iy)| = |2x + 2iy| = 2\sqrt{x^2 + y^2} \]

**Hint:** The vertices of a square are equidistant from each other.

### Step 7: Set the side length equal to 4
Since the side length of the square is given as 4, we have:
\[ 2|y| = 4 \implies |y| = 2 \]
\[ 2\sqrt{x^2 + y^2} = 4 \implies \sqrt{x^2 + y^2} = 2 \implies x^2 + y^2 = 4 \]

**Hint:** Use the Pythagorean theorem to relate the sides of the square to the coordinates.

### Step 8: Substitute \( |y| \)
Since \( |y| = 2 \), we can substitute this into the equation:
\[ x^2 + 2^2 = 4 \implies x^2 + 4 = 4 \implies x^2 = 0 \implies x = 0 \]

**Hint:** This indicates that the real part of \( z \) is zero.

### Step 9: Find \( |z| \)
Now we have \( z = 0 + 2i \) or \( z = 0 - 2i \). The modulus of \( z \) is:
\[ |z| = \sqrt{x^2 + y^2} = \sqrt{0^2 + 2^2} = \sqrt{4} = 2 \]

**Hint:** The modulus of a complex number is the distance from the origin in the Argand plane.

### Final Answer
Thus, the value of \( |z| \) is \( 2 \).