Complex Numbers
Geometric Interpretation
Grade 11
Question:
<p>The triangle formed by the complex numbers \(z\), \(iz\), \(i^2z\) is:</p>
<p>(a) equilateral</p>
<p>(b) isosceles</p>
<p>(c) right angled</p>
<p>(d) isosceles but not right angled</p>
Step-by-Step Solution
Key Concept: Multiplication by $i$ rotates a complex number by $90°$ about the origin; use distance formula to find side lengths.
<p><strong>Step 1:</strong> Identify the three vertices: $A = z$, $B = iz$, $C = i^2z = -z$.</p><p><strong>Step 2:</strong> Calculate the distances (side lengths):</p><p>$|AB| = |iz - z| = |z||i - 1| = |z|\sqrt{2}$</p><p>$|BC| = |-z - iz| = |z||-1 - i| = |z|\sqrt{2}$</p><p>$|CA| = |z - (-z)| = |2z| = 2|z|$</p><p><strong>Step 3:</strong> Check if right-angled using $|CA|^2 = |AB|^2 + |BC|^2$:</p><p>$(2|z|)^2 = (|z|\sqrt{2})^2 + (|z|\sqrt{2})^2$</p><p>$4|z|^2 = 2|z|^2 + 2|z|^2 = 4|z|^2$ ✓</p><p>∴ The triangle is right-angled isosceles with the right angle at $B$.</p>
Correct Answer: c