Complex Numbers
Geometric interpretation
Grade 11
Question:
<p>\(z_1, z_2, z_3, z_4\) are distinct complex numbers representing the vertices of a quadrilateral \(ABCD\) taken in order. If \(z_1 - z_4 = z_2 - z_3\) and \(\arg[(z_4 - z_1)/(z_2 - z_1)] = \pi/2\), then the quadrilateral is</p>
<p>rectangle</p>
<p>rhombus</p>
<p>square</p>
<p>trapezium</p>
Step-by-Step Solution
Key Concept: The condition z₁ - z₄ = z₂ - z₃ means opposite sides are equal (parallelogram property), and arg[(z₄ - z₁)/(z₂ - z₁)] = π/2 means the sides are perpendicular, making it a rectangle.
<p><strong>Step 1:</strong> Interpret z₁ - z₄ = z₂ - z₃</p><p>Rearranging: z₁ + z₃ = z₂ + z₄</p><p>This means the diagonals bisect each other (midpoint of AC = midpoint of BD), so ABCD is a parallelogram.</p><p><strong>Step 2:</strong> Interpret arg[(z₄ - z₁)/(z₂ - z₁)] = π/2</p><p>Let the ratio equal w where arg(w) = π/2, so w = ki for some real k > 0</p><p>This means: z₄ - z₁ = ki(z₂ - z₁), which shows vector AB ⊥ vector AD</p><p><strong>Step 3:</strong> Conclusion</p><p>A parallelogram with perpendicular adjacent sides is a rectangle.</p><p>∴ Answer: C (Rectangle)</p>
Correct Answer: C