Complex Numbers
Equilateral triangle in complex plane
Grade 11
Question:
<p>If \(z_1, z_2, z_3\) are the vertices of an equilateral triangle \(ABC\) such that \(|z_1 - i| = |z_2 - i| = |z_3 - i|\), then \(|z_1 + z_2 + z_3|\) equals</p>
<p>\(3\sqrt{3}\)</p>
<p>\(\sqrt{3}\)</p>
<p>\(3\)</p>
<p>\(\frac{1}{3\sqrt{3}}\)</p>
Step-by-Step Solution
Key Concept: If three points are equidistant from a single point, that point is the circumcenter of the triangle. For an equilateral triangle, the circumcenter coincides with the centroid, so i is the centroid.
<p><strong>Step 1:</strong> The condition |z₁ - i| = |z₂ - i| = |z₃ - i| means all three vertices are equidistant from the point i. Therefore, i is the circumcenter of triangle ABC.</p><p><strong>Step 2:</strong> For an equilateral triangle, the circumcenter coincides with the centroid. The centroid is given by: i = (z₁ + z₂ + z₃)/3</p><p><strong>Step 3:</strong> Solving for the sum: z₁ + z₂ + z₃ = 3i</p><p><strong>Step 4:</strong> Taking the modulus: |z₁ + z₂ + z₃| = |3i| = 3|i| = 3(1) = 3</p><p>∴ Answer: C (which equals 3)</p>
Correct Answer: C