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 (1, 2), (4, y), (x, 6), and (3, 5) are the coordinates of A, B, C, and D vertices of a parallelogram ABCD.
Intersection point O of diagonal AC and BD also divides these diagonals.
Therefore, O is the mid-point of AC and BD.

If O is the mid-point of AC, then the coordinates of O are
(\(\frac{1+x}{2},\frac{2+6}{2}\Rightarrow (\frac{x+1}{2},4)\)
If O is the mid-point of BD, then the coordinates of O are 
\((\frac{4+3}{2},\frac{5+y}{2})\Rightarrow (\frac{7}{2},\frac{5+y}{2})\)

Since both the coordinates are of the same point O,
\(\therefore\)\(\frac{x+1}{2}=\frac{7}{2}\, \text{ and }\,4=\frac{5+y}{2}\)
\(\Rightarrow x+1=7\, \text{ and }\, 5+y=8\)
\(\Rightarrow\,x=6\, \text{ and }\,y=3\)