Question

If $a,b,c$ and $u,v,w$ are 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

• A. have the same area
• B. are similar
• C. are congruent
• D. none of these

Answer 1

The correct option is B are similar

Let $A\left(a\right),B\left(b\right),C\left(c\right)$ and $P\left(u\right),Q\left(v\right),R\left(w\right)$


Now, $c=(1-r)a+rb$ and $w=(1-r)u+rv$
$\Rightarrow \frac{c-a}{b-a}=\frac{w-u}{v-u}=r$
$\Rightarrow \left|\frac{c-a}{b-a}\right|=\left|\frac{w-u}{v-u}\right|=|r|$
$\Rightarrow \frac{AC}{AB}=\frac{PR}{PQ}=|r|\text{and}\angle CAB=\angle RPQ$

Hence, both the triangles are similar.