<p>If center of a regular hexagon is at the origin and one of the vertices on the Argand diagram is \(1 + 2i\), then its perimeter is</p>
Step-by-Step Solution
Key Concept: In a regular hexagon centered at the origin, all vertices are equidistant from the origin. This distance equals the side length of the hexagon, so the modulus of any vertex gives the side length directly.
<p><strong>Step 1:</strong> For a regular hexagon centered at the origin, the vertices lie on a circle of radius r (circumradius). The key property is that in a regular hexagon, the side length equals the circumradius.</p><p><strong>Step 2:</strong> Find the circumradius using the given vertex z = 1 + 2i:</p><p>r = |1 + 2i| = √(1² + 2²) = √(1 + 4) = √5</p><p><strong>Step 3:</strong> Since the side length of a regular hexagon equals its circumradius:</p><p>Side length = √5</p><p><strong>Step 4:</strong> A regular hexagon has 6 sides, so:</p><p>Perimeter = 6 × √5 = 6√5</p><p>∴ Answer: D</p>
Correct Answer: D