Question

$A B C D$ is a rhombus such that its diagonals $A C$ and $BD$ intersect at point $M$ and satisfy $BD =2 AC$. If points $D$ and $M$ represent the complex numbers $1+i$ and $2-i$ respectively then find $[S]$ where $S$ is the area of $\square A B C D$ and $[ \,\,\, ]$ represents the greatest integer.

Answer 1

Vector $MD$ represents $1+i-(2-i)=-1+2 i$
Vector $MA$ represents $=\frac{1}{2}[i(-1+2 i)] \ldots\left[| MA |=\frac{1}{2}| DM |\right]$
$=-\frac{i}{2}-1$
$\Rightarrow| MA |=\sqrt{\frac{1}{4}+1}=\frac{\sqrt{5}}{2} $
$\Rightarrow| AC |=\sqrt{5}$
$| DM |=\sqrt{(-1)^{2}+2^{2}}=\sqrt{5}$
$ \Rightarrow| BD |=2 \sqrt{5}$
$\Rightarrow$ Area of $\square ABCD =\frac{1}{2}| AC || BD |$
i.e. $S =\frac{1}{2}(\sqrt{5})(2 \sqrt{5})$
$\Leftrightarrow S =5$
$\Rightarrow[S]=5$