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 thses

Answer 1

The correct option is B Are similar

Let, $a,b,c$ and $u,v,w$ be complex numbers denoting the vertices $A,B,C$ of $\Delta$ABC and $U,V,W$ of $\Delta$UVW respectively.

Given, $c=(1-r)a+rb$.

Now, for the $\Delta$ ABC the length of the sides

$AB=\overline{AB}=|b-a|$,

$\overline{BC}=|r||(b-a)|$ and $\overline{CA}=|(1-r)||b-a|$.

Now, for $\Delta$ UVW the length of the sides

Also given $w=(1-r)u+rv$.

$\overline{UV}=|v-u|$,

$\overline{VW}=|r||(v-u)|$ and $\overline{WU}=|(1-r)||v-u|$.

Now, we have

$\frac{\overline{AB}}{\overline{UV}}=\frac{|b-a|}{|v-u|}$,$\frac{\overline{BC}}{\overline{VW}}=\frac{|b-a|}{|v-u|}$ and $\frac{\overline{CA}}{\overline{WU}}=\frac{|b-a|}{|v-u|}$.

So, $\frac{\overline{AB}}{\overline{UV}}=$$\frac{\overline{BC}}{\overline{VW}}=$$\frac{\overline{CA}}{\overline{WU}}$.

So, the triangles are similar.