Find co-ordinates of A.

Question

$ABCD$ is a rhombus . Diagonals $AC$ and $BD$ intersect at $M$.
$BD=2AC$
$D\rightarrow(1,1)$ and $M\rightarrow(2,-1)$

Find the co-ordinates of $A$.

Answer 1

For $DM$ we obtain: $$m_{DM}=\frac{1+1}{1-2}=-2,$$ which says the equation of $AC$ it's $$y+1=\frac{1}{2}(x-2)$$ or $$y=\frac{1}{2}x-2.$$ Now, let $A\left(x,\frac{1}{2}x-2\right)$.

Thus, since $$AM=\frac{1}{2}DM=\frac{1}{2}\sqrt{(1-2)^2+1+1)^2}=\frac{1}{2}\sqrt5,$$ we obtain: $$(x-2)^2+\left(\frac{1}{2}x-1\right)^2=\frac{5}{4}$$ or $$(x-2)^2=1,$$ which gives $x=3$ or $x=1$ and we obtain: $$A\left(3,-\frac{1}{2}\right)$$ or $$A\left(1,-\frac{3}{2}\right).$$