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 no.

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

Answer 1

The correct option is B are similar

Since $a,b,c$ and $u,v,w$ are the vertices of two triangles
also $c=(1-r)a+rb$
and $w=(1-r)u+rv$ ...(1)
consider $\left|\begin{array}{ccc}a& u& 1\\ b& v& 1\\ c& w& 1\end{array}\right|$
Applying $R_3\to R_3-\{(1-r)R_1-{rR}_2\}$
$R_3\to R_3-\{(1-r)R_1-{rR}_2\}=\left|\begin{array}{ccc}a& u& 1\\ b& v& 1\\ c-(1-r)a-rb& w-(1-r)u-eb& 1-(1-r)-r\end{array}\right|=\left|\begin{array}{ccc}a& u& 1\\ b& v& 1\\ 0& 0& 0\end{array}\right|=0$
Hence, two triangles are similar