Question

If $A\left(a\right),B\left(b\right),C\left(c\right)$ and $P\left(u\right),Q\left(v\right),R\left(w\right)$ are complex numbers representing the vertices of two triangles such that $△ABC$ and $△PQR$ are similar, then $\frac{a-c}{a-b}=$

• A. $\frac{u-v}{u-w}$
• B. $\frac{u-v}{v-w}$
• C. $\frac{u-w}{u-v}$
• D. $\frac{u-w}{w-v}$

Answer 1

The correct option is C $\frac{u-w}{u-v}$


In the figure triangles are similar. Hence
$\frac{AC}{AB}=\frac{PR}{PQ}$
$\Rightarrow \left|\frac{a-c}{a-b}\right|=\left|\frac{u-w}{u-v}\right|\cdots \left(1\right)$
Now
$\frac{a-c}{a-b}=\left|\frac{a-c}{a-b}\right|e^{i\theta }$
$=\left|\frac{u-w}{u-v}\right|e^{i\theta }\left[\text{using}\right(1\left)\right]=\frac{u-w}{u-v}[\because \angle BAC=\angle QPR]$