Question

$ABCD$ is a rhombus. Its diagonals $AC$ and $BD$ intersect at $M$ such that $BD=2AC$. If the points $D$ and $M$ represent the complex number $1+i$ and $2–i$ respectively, find the complex number(s) representing $A$.

Answer 1

Let $A$ be $z$. The position $MA$ can be obtained by rotating $MD$ anticlockwise through an angle $\frac{\pi }{2}$ ; simultaneously length gets halved.

$\therefore z-(2-i)=\frac{1}{2}((1+i)-(2-i))e^{\frac{i\pi }{2}}$

$=\frac{1}{2}(-2-i)$

$=-1-\frac{1}{2}i$

$z=-1-\frac{1}{2}i+2-i=1-\frac{3i}{2}$

Another position of $A$ corresponds to $A$ and $C$ getting interchanged and in that the complex number of $A$ is $1+\frac{1}{2}i+2-i=3-\frac{1}{2}i$

$\therefore$The complex number of $A$ is either $1-\frac{3i}{2}$ or $3-\frac{i}{2}$