Complex Numbers
Geometry of complex numbers
Grade 11
Question:
<p>\(P(z_1)\), \(Q(z_2)\), \(R(z_3)\), and \(S(z_4)\) are four complex numbers representing the vertices of a rhombus taken in order on the complex plane, then which one of the following is/are correct?</p>
<p>\(\dfrac{z_1 - z_4}{z_2 - z_3}\) is purely real</p>
<p>\(\text{amp}\dfrac{z_1 - z_4}{z_2 - z_4} = \text{amp}\dfrac{z_2 - z_4}{z_3 - z_4}\)</p>
<p>\(\dfrac{z_1 - z_3}{z_2 - z_4}\) is purely imaginary</p>
<p>it is not necessary that \(|z_1 - z_3| \neq |z_2 - z_4|\)</p>
Step-by-Step Solution
Key Concept: In a rhombus, the diagonals bisect each other. Therefore, the midpoint of diagonal PR equals the midpoint of diagonal QS, giving z₁ + z₃ = z₂ + z₄. Additionally, all four sides are equal: |z₂ - z₁| = |z₃ - z₂| = |z₄ - z₃| = |z₁ - z₄|.
<p><strong>Step 1:</strong> For a rhombus PQRS with vertices in order, the diagonals PR and QS bisect each other at their midpoint.</p><p><strong>Step 2:</strong> Midpoint of PR = (z₁ + z₃)/2 and Midpoint of QS = (z₂ + z₄)/2. Setting these equal: (z₁ + z₃)/2 = (z₂ + z₄)/2</p><p><strong>Step 3:</strong> This simplifies to: <strong>z₁ + z₃ = z₂ + z₄</strong></p><p><strong>Step 4:</strong> All sides of a rhombus are equal in length, so: |z₂ - z₁| = |z₃ - z₂| = |z₄ - z₃| = |z₁ - z₄|</p><p><strong>Step 5:</strong> The diagonals of a rhombus bisect each other at right angles (perpendicular), meaning: (z₃ - z₁)/(z₄ - z₂) is purely imaginary or their dot product is zero.</p><p>∴ Answer: ABC (all three statements about diagonal bisection and equal sides are correct)</p>
Correct Answer: ABC