Question

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:

• A. \( \frac{3\sqrt{3}}{4} \)
• B. \( \frac{\sqrt{3}}{4} \)
• C. \( 1 \)
• D. \( 2 \)

Hint:

Complex numbers with equal magnitude and a sum of zero, when representing vertices of a polygon, often form regular polygons centered at the origin.

Answer 1

The Correct Option is A

Step 1: Geometric Interpretation.
The conditions \( |z_1| = |z_2| = |z_3| = 1 \) imply the vertices lie on the unit circle. \( z_1 + z_2 + z_3 = 0 \) implies the centroid of the triangle is at the origin.
Step 2: Type of Triangle.
A triangle with vertices on a circle and centroid at the center must be equilateral.
Step 3: Side Length of the Triangle.
Let \( z_k = e^{i\theta_k} \) for \( k = 1, 2, 3 \). The angles between the vertices are \( \frac{2\pi}{3} \). The side length \( a \) is: \[ a = |z_1 - z_2| = |e^{i\alpha} - e^{i(\alpha + \frac{2\pi}{3})}| = \sqrt{3} \]
Step 4: Area of the Equilateral Triangle.
The area \( A \) of an equilateral triangle with side \( a \) is \( A = \frac{\sqrt{3}}{4} a^2 \). \[ A = \frac{\sqrt{3}}{4} (\sqrt{3})^2 = \frac{3\sqrt{3}}{4} \]