Question

Show that the points representing the complex numbers $7+9i$, $-3+7i$, $3+3i$ form a right angled triangle on the Argand diagram.

Answer 1

Let $A,B,C$ represent the complex numbers $7+9i,-3+7i,3+3i$ in the Argand diagram respectively.

$AB=|(7+9i)-(-3+7i)|$

$=|10+2i|=\sqrt{{10}^2+2^2}=\sqrt{104}$

$BC=|(-3+7i)-(3+3i)|$

$=|-6+4i|=\sqrt{(-6{)}^2+4^2}=\sqrt{52}$

$CA=|(3+3i)-(7+9i)|$

$=|-4-6i|=\sqrt{(-4{)}^2+(-6{)}^2}=\sqrt{52}$

$\Rightarrow A{B}^2=B{C}^2+C{A}^2$

$\Rightarrow \angle BCA={90}^{\circ }$ and $BC=CA$

Hence $△ABC$ is a right angled isosceles triangle.