Complex Numbers
Rotation – Type of Triangle from Complex Ratio
Complex Numbers_PYQ
Grade 11
Question:
The complex numbers $z_1, z_2$ and $z_3$ satisfying $\dfrac{z_1-z_3}{z_2-z_3}=\dfrac{1-i\sqrt{3}}{2}$ are the vertices of a triangle which is
of area zero
right angled isosceles
equilateral
obtuse angled isosceles
Step-by-Step Solution
Key Concept: The complex ratio $(z_1-z_3)/(z_2-z_3)$ encodes both the ratio of side lengths ($=|\text{ratio}|$) and the angle at $z_3$ ($=|\arg(\text{ratio})|$). Modulus $1$ and angle $60°$ forces equilateral.
**Step 1: Find modulus of the ratio**
$\left|\dfrac{1-i\sqrt{3}}{2}\right|=\dfrac{\sqrt{1+3}}{2}=1$, so $|z_1-z_3|=|z_2-z_3|$. Two sides of the triangle are equal (isosceles from $z_3$).
**Step 2: Find argument of the ratio**
$\arg\!\left(\dfrac{1-i\sqrt{3}}{2}\right)=\arg(1-i\sqrt{3})=-\dfrac{\pi}{3}$. So the angle at vertex $z_3$ is $\dfrac{\pi}{3}=60°$.
**Step 3: Conclude triangle type**
By the law of cosines with $|z_1-z_3|=|z_2-z_3|=r$ and included angle $\pi/3$: $|z_1-z_2|^2=2r^2-2r^2\cos(\pi/3)=r^2$, so $|z_1-z_2|=r$. All three sides equal $\Rightarrow$ equilateral.
Correct Answer: 3