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

Answer: (B)

Are similar

$c=(1-r)a+rb$ implies $\frac{c-a}{b-a}=r$. Similarly, $w=(1-r)u+rv$ implies $\frac{w-u}{v-u}=r$. Thus,$\frac{\left(AC\right)}{\left(AB\right)}=\frac{\left(PR\right)}{\left(PQ\right)}$ and $\angle \left(BAC\right)=\angle \left(QPR\right)$. This implies that triangle $ABC$ and triangle $PQR$ are similar triangles . Hence, option B is correct.