If $(1,2),(4,y),(x,6)$ and $(3,5)$ are the vertices of parallelogram taken in order, find $x$ and $y$.

The correct option is A. $(6,3)$

The given points are: $A(1,2),B(4,y),C(x,6),$ and $D(3,5)$

We know that the diagonals of a parallelogram bisect each other.

Also, we know that mid-point of the line joining two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Mid point of $AC=(\frac{1+x}{2},\frac{2+6}{2})=(\frac{1+x}{2},4)$

Mid point of $BD=(\frac{4+3}{2},\frac{5+y}{2})=(\frac{7}{2},\frac{5+y}{2})$

Now,

Mid point of $AC=$Mid point of $BD$

$\Rightarrow (\frac{1+x}{2},4)=(\frac{7}{2},\frac{5+y}{2})$

$\Rightarrow \frac{1+x}{2}=\frac{7}{2},4=\frac{5+y}{2}$

$\Rightarrow 1+x=7,5+y=8$

$\Rightarrow x=7-1,y=8-5$

$\therefore x=6,y=3$.