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

Let A (1, 2), B (4, y), C(x, 6), and D (3, 5) be the vertices of a parallelogram ABCD.

Since the diagonals of a parallelogram bisect each other. The intersection point O of diagonal AC and BD also divides these diagonals in the ratio 1:1.

Therefore, O is the mid-point of AC and BD.

According to the midpoint formula,

O(x, y) = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]

If O is the mid-point of AC, then the coordinates of O are

[(1 + x) / 2, (2 + 6) / 2]

⇒ [(x + 1) / 2, 4] ----- (1)

If O is the mid-point of BD, then the coordinates of O are

[(4 + 3) / 2, (5 + y) / 2]

⇒ [7/2, (5 + y) / 2] ------ (2)

Since both the coordinates are of the same point O, so, (x + 1) / 2 = 7 / 2 and 4 = (5 + y) / 2 [From equation(1) and (2)]

⇒ x + 1 = 7 and 5 + y = 8  (By cross multiplying & transposing)

⇒ x = 6 and y = 3