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) are the vertices of a parallelogram..

We know that,

Diagonals of parallelogram bisect each other.

So, mid-point will be same let O.

i.e., Mid-point of AC = Mid-point of BD.

By formula,

Mid-point = $\Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

Substituting values we get :

$\Rightarrow \Big(\dfrac{1 + x}{2}, \dfrac{2 + 6}{2}\Big) = \Big(\dfrac{4 + 3}{2}, \dfrac{y + 5}{2}\Big) \\[1em] \Rightarrow \Big(\dfrac{1 + x}{2}, \dfrac{8}{2}\Big) = \Big(\dfrac{7}{2}, \dfrac{y + 5}{2}\Big) \\[1em] \Rightarrow \dfrac{1 + x}{2} = \dfrac{7}{2} \text{ and } \dfrac{8}{2} = \dfrac{y + 5}{2} \\[1em] \Rightarrow 1 + x = 7 \text{ and } 8 = y + 5 \\[1em] \Rightarrow x = 7 - 1 \text{ and } y = 8 - 5 \\[1em] \Rightarrow x = 6 \text{ and } y = 3.$

Hence, x = 6 and y = 3.