• # question_answer 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

Let the complex number $a,b,c$and $u,v,w$ represent the vertices $A,B,C$and $D,E,F$ of the two triangle $ABC$ and $DEF$ respectively. Put  $b-a={{r}_{1}}{{e}^{i{{\theta }_{1}}}}$      $c-a={{r}_{2}}{{e}^{i{{\theta }_{2}}}}$         $v-u={{\rho }_{1}}{{e}^{i{{\varphi }_{1}}}},w-u={{\rho }_{2}}{{e}^{i{{\varphi }_{2}}}}$and $r=\lambda {{e}^{i\alpha }}$ Substituting these values in the given relations $c-a=r(b-a)$and $w-u=(v-u)r,$ we have         ${{r}_{2}}{{e}^{i{{\theta }_{2}}}}=\lambda {{e}^{i\alpha }}{{r}_{1}}{{e}^{i{{\theta }_{1}}}}=\lambda {{r}_{1}}{{e}^{i(\alpha +{{\theta }_{1}})}}$            .......(i) and ${{\rho }_{2}}{{e}^{i{{\varphi }_{2}}}}={{\rho }_{1}}{{e}^{i{{\varphi }_{1}}}}\lambda {{e}^{i\alpha }}=(\lambda {{\rho }_{1}}){{e}^{i({{\varphi }_{1}}+\alpha )}}$       .......(ii) Equating moduli and arguments of the complex numbers on both sides (i), we get ${{r}_{2}}=\lambda {{r}_{1}},{{\theta }_{2}}=\alpha +{{\theta }_{1}}$ i.e., $AC=\lambda AB$and $\angle CAB={{\theta }_{2}}-{{\theta }_{1}}=\alpha$ Similarly from (ii), we shall get $DF=\lambda DE$ and $\angle FDE={{\varphi }_{2}}-{{\varphi }_{1}}=\alpha$ Thus we get $\frac{AC}{DF}=\frac{AB}{DE}$and $\angle CAB=\angle FDE$ Hence the triangle $ABC$ and $DEF$ are similar.