Question

Consider the following two statements.

Statement I : If $z_1 + \omega z_2 + \omega^2 z_3 = 0$, where $\omega$ and $\omega^2$ are the non-real complex cube roots of unity, then $z_1, z_2$ and $z_3$ are the vertices of an equilateral triangle.

Statement II : If $z_3 - z_1 = (z_2 - z_1)e^{i\frac{\pi}{3}}$, then $z_1, z_2$ and $z_3$ are vertices of an equilateral triangle.

• A. Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I
• B. Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I
• C. Statement I is true; Statement II is false
• D. Statement I is false; Statement II is true

Answer 1

Answer: (B)

Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I

Answer 2

Statement I Analysis: The condition for $z_1, z_2, z_3$ to form an equilateral triangle is $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$. This equation can be factored as $(z_1 + \omega z_2 + \omega^2 z_3)(z_1 + \omega^2 z_2 + \omega z_3) = 0$. Thus, $z_1, z_2, z_3$ form an equilateral triangle if and only if $z_1 + \omega z_2 + \omega^2 z_3 = 0$ or $z_1 + \omega^2 z_2 + \omega z_3 = 0$. Statement I is true because $z_1 + \omega z_2 + \omega^2 z_3 = 0$ is one of the conditions for forming an equilateral triangle.

Statement II Analysis: The condition $z_3 - z_1 = (z_2 - z_1)e^{i\frac{\pi}{3}}$ implies that the vector from $z_1$ to $z_3$ is obtained by rotating the vector from $z_1$ to $z_2$ by $\frac{\pi}{3}$ (60 degrees). This means $|z_3 - z_1| = |z_2 - z_1|$ (sides are equal) and the angle between these sides at $z_1$ is $\frac{\pi}{3}$. A triangle with two equal sides and an included angle of $60^\circ$ is equilateral. Statement II is true.

Relationship Analysis: Statement I provides an algebraic condition, and Statement II provides a geometric condition for an equilateral triangle. While both statements are true, Statement II does not explain Statement I. Statement II describes a specific orientation of an equilateral triangle (with a 60-degree angle at $z_1$), whereas Statement I's condition ($z_1 + \omega z_2 + \omega^2 z_3 = 0$) is a more general algebraic condition that implies an equilateral triangle, and it does not directly translate to the geometric condition in Statement II. For example, if $z_1=1, z_2=\omega, z_3=\omega^2$, Statement I is true, but Statement II is not satisfied by these points.