Complex Numbers
Argument – Similarity of Triangles via Complex Ratio
Complex Numbers_PYQ
Grade 11
Question:
If $a,b,c$ and $u,v,w$ are the complex numbers representing the vertices of two triangles such that $c=(1-r)a+rb$ and $w=(1-r)u+rv$, where $r$ is a complex number, then the two triangles
have the same area
are similar
are congruent
None of these
Step-by-Step Solution
Key Concept: Two triangles are similar iff the complex ratio of corresponding side vectors from one vertex is equal. The ratio $r=(c-a)/(b-a)=(w-u)/(v-u)$ encodes both the length ratio $|r|$ and the rotation angle $\arg(r)$.
**Step 1: Extract the complex ratio from first triangle**
$c=(1-r)a+rb \Rightarrow c-a=r(b-a) \Rightarrow r=\dfrac{c-a}{b-a}$.
**Step 2: Same ratio for second triangle**
$w-u=r(v-u) \Rightarrow \dfrac{w-u}{v-u}=r=\dfrac{c-a}{b-a}$.
**Step 3: Conclude similarity**
Equal complex ratios $\dfrac{c-a}{b-a}=\dfrac{w-u}{v-u}$ means the triangles share the same angle at $a$ (resp. $u$) and the ratio of the adjacent sides is $|r|$. The two triangles are similar.
Correct Answer: 2