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 Understanding.
The constraints \( |z_1| = |z_2| = |z_3| = 1 \) mean the vertices are on the unit circle. \( z_1 + z_2 + z_3 = 0 \) indicates the triangle's centroid is at the origin.

Step 2: Triangle Classification.
A triangle with vertices on a circle and its centroid at the center must be equilateral.

Step 3: Triangle Side Length.
Let \( z_k = e^{i\theta_k} \) for \( k = 1, 2, 3 \). The angles between vertices are \( \frac{2\pi}{3} \). The side length \( a \) is:\n\[\na = |z_1 - z_2| = |e^{i\alpha} - e^{i(\alpha + \frac{2\pi}{3})}| = \sqrt{3}\n\]
Step 4: Equilateral Triangle Area.
The area \( A \) of an equilateral triangle with side \( a \) is \( A = \frac{\sqrt{3}}{4} a^2 \).\n\[\nA = \frac{\sqrt{3}}{4} (\sqrt{3})^2 = \frac{3\sqrt{3}}{4}\n\]