Question

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

Answer 1

Answer: 5

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