Question

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

Answer 1

Let A,B,C and D be the points (1,2) (4,y), (x,6) and (3,5) respectively.
Mid point of diagonal AC is $(\frac{1+x}{2},\frac{2+6}{2})\Rightarrow (\frac{x+1}{2},4)$
Mid point of diagonal BD is$(\frac{4+3}{2},\frac{5+y}{2})\Rightarrow (\frac{7}{2},\frac{5+y}{2})$
Since the diagonals of a parallelogram bisect each other, the midpoints of AC and BD are the same.
$\therefore \frac{x+1}{2}=\frac{7}{2}$ and $4=\frac{5+y}{2}$
$\Rightarrow x+1=7$ and $5+y=8$
$\Rightarrow x=6$ and $y=3$