Complex Numbers
Geometric Representation
Grade 11
Question:
<p>If centre of a regular hexagon is at origin and one of the vertex on argand diagram is <math>1 + 2i</math>, where <i>i</i> = <math>\sqrt{-1}</math>, its perimeter is</p>
<p>(a) <math>2\sqrt{5}</math></p>
<p>(b) <math>6\sqrt{2}</math></p>
<p>(c) <math>4\sqrt{5}</math></p>
<p>(d) <math>6\sqrt{5}</math></p>
Step-by-Step Solution
Key Concept: In a regular hexagon centered at the origin, all vertices are equidistant from the center. The distance from the center to any vertex equals the side length of the hexagon. Calculate this distance using the modulus of the complex number representing the vertex.
<p><strong>Step 1:</strong> Identify the vertex in the Argand diagram. One vertex is at z = 1 + 2i.</p><p><strong>Step 2:</strong> Find the distance from the center (origin) to this vertex using the modulus formula:</p><p>|z| = |1 + 2i| = √(1² + 2²) = √(1 + 4) = √5</p><p><strong>Step 3:</strong> Recognize the key property of a regular hexagon: the distance from the center to any vertex equals the side length of the hexagon. This is because a regular hexagon can be divided into 6 equilateral triangles, each with vertices at the center and two adjacent vertices of the hexagon.</p><p><strong>Step 4:</strong> Therefore, the side length of the hexagon is √5.</p><p><strong>Step 5:</strong> Calculate the perimeter by multiplying the side length by the number of sides:</p><p>Perimeter = 6 × side length = 6√5</p><p><strong>∴ Answer:</strong> D</p>
Correct Answer: D