Key Concepts and Formulas

• Complex Number Representation: $z = x + iy$, where $x = \text{Re}(z)$ and $y = \text{Im}(z)$.
• Conjugate: $\overline{z} = x - iy$.
• Modulus: $|z| = \sqrt{x^2 + y^2}$. Distance between $z_1$ and $z_2$ is $|z_1 - z_2|$.

Step-by-Step Solution

Step 1: Represent the complex number and its conjugate. Let $z = x + iy$, where $x$ and $y$ are real numbers. Then $\overline{z} = x - iy$. Explanation: This sets up the foundation by expressing the complex number in terms of its real and imaginary parts and defining its conjugate.*

Step 2: Express the vertices of the square in terms of $x$ and $y$. The given vertices are: $V_1 = z = x + iy$ $V_2 = \overline{z} = x - iy$ $V_3 = \overline{z} - 2\text{Re}(\overline{z}) = (x - iy) - 2x = -x - iy$ $V_4 = z - 2\text{Re}(z) = (x + iy) - 2x = -x + iy$ Explanation: This step simplifies the expressions for each vertex, making it easier to visualize their positions in the Argand plane.*

Step 3: Calculate the side length using the distance between adjacent vertices. The side length of the square is 4. We can use the distance between $V_1$ and $V_2$ to find a relationship between $x$ and $y$.

$|V_1 - V_2| = |(x + iy) - (x - iy)| = |2iy| = 2|y| = 4$ Therefore, $|y| = 2$.

Similarly, we can use the distance between $V_1$ and $V_4$: $|V_1 - V_4| = |(x + iy) - (-x + iy)| = |2x| = 2|x| = 4$ Therefore, $|x| = 2$. Explanation: This step uses the given side length of the square to establish the magnitudes of the real and imaginary parts of $z$.*

Step 4: Calculate the modulus of $z$. $|z| = \sqrt{x^2 + y^2} = \sqrt{(\pm 2)^2 + (\pm 2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$ Explanation: This step applies the formula for the modulus of a complex number, using the values found in the previous step.*

Common Mistakes & Tips

• Sign Errors: Be careful with signs when substituting and simplifying expressions.
• Modulus Definition: Remember that $|z| = \sqrt{x^2 + y^2}$ and not $x+y$.
• Geometric Visualization: Sketching the points in the Argand plane can help in understanding the relationships and avoiding errors.

Summary

We expressed the vertices of the square in terms of the real and imaginary parts of $z$. By calculating the distance between adjacent vertices and equating it to the given side length, we found that $|x|=2$ and $|y|=2$. Finally, we calculated the modulus of $z$ as $|z| = \sqrt{x^2 + y^2} = 2\sqrt{2}$.

Final Answer The final answer is \boxed{2\sqrt{2}}, which corresponds to option (D).