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 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 the triangles be $△ABC$ and $△UWV$.

Vertices of $△ABC$ be represented by $a,b,c$ and $△UWV$ be represented by $u,v,w$.

$AB=b-a$

$BC=c-b=(1-r)a+rb-b=a-b-r(a-b)=(a-b)(1-r)$

$CA=a-c=a-(1+r)a-rb=r(a-b)$

$UV=v-u$

$VW=w-v=(1-r)u+rv-v=u-v-r(u-v)=(u-v)(1-r)$

$WU=u-w=u-(1-r)u-rv=r(u-v)$

$\frac{AB}{UV}=\frac{b-a}{v-u}=\frac{a-b}{u-v}$ ------------------(1)

$\frac{BC}{VW}=\frac{(a-b)(1-r)}{(u-v)(1-r)}=\frac{a-b}{u-v}$----(2)

$\frac{CA}{WU}=\frac{r(a-b)}{r(u-v)}=\frac{a-b}{u-v}$ ------------(3)

From (1),(2) and (3),

$\frac{AB}{UV}=\frac{BC}{VW}=\frac{CA}{WU}$

$\therefore △ABC\sim △UVW$