Question

If the complex numbers $iz$, $z$ and $z+iz$ represents the three vertices of a triangle, then the area of the triangle is

• A. $\frac{1}{2}\left|z-1\right|$
• B. ${\left|z\right|}^2$
• C. $\frac{1}{2}{\left|z\right|}^2$
• D. ${\left|z-1\right|}^2$

Answer 1

The correct option is C $\frac{1}{2}{\left|z\right|}^2$

Let $z=x+iy,iz=-y+ix,z+iz=x-y+i(x+y)$

vertices are $(x,y);(-y,x);(x-y,x+y)$

Area$=\frac{1}{2}\left|\begin{array}{ccc}x& -y& x-y\\ y& x& x+y\\ 1& 1& 1\end{array}\right|=\frac{1}{2}|x^2+y^2|=\frac{1}{2}{\left|z\right|}^2$