Question

The triangle with vertices at the points $z_1,z_2,(1-i)z_1+i{z}_2$ is?

• A. Right angled bit not isosceles
• B. Isosceles but not right angled
• C. Right angled and isosceles
• D. Equilateral

Answer 1

Answer: (C)

Right angled and isosceles

Let $A=z_1,B=z_2,C=(1-i)z_1+i{z}_2$ be the vertices of triangle ABC

$\Rightarrow \left|AB\right|=|z_1-z_2|$

$\Rightarrow \left|BC\right|=|(1-i)z_1+i{z}_1-z_2|=\left|(1-i)(z_1-z_2)\right|$

$=|1-i||z_1-z_2|$

$=\sqrt{2}|z_1-z_2|$

$\Rightarrow \left|AC\right|=|z_1+(1-i)z_1+i{z}_2|=|-i(z_1-z_2)|$

$=|-i||z_1-z_2|$

$=|z_1-z_2|$

$\Rightarrow \left|AB\right|=\left|AC\right|$

$\&\left|BC\right|=\sqrt{2}\left|AB\right|=\sqrt{2}\left|AC\right|$

$\therefore ABC$ is right isosceles triangle.

Hence, solved.