#### Question

If the points (-2, -1), (1, 0), (x, 3) and (1, y) form a parallelogram, find the values of x and y.

#### Solution

Let ABCD be a parallelogram in which the coordinates of the vertices are A (−2,−1); B (1, 0); C (*x*, 3) and D (1, *y*).

Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.

In general to find the mid-point `P(x,y)` of two points `A(x_1, y_1)` and `B(x_2, y_2)` we use section formula as,

`P(x, y) = ((x_1+ x_2)/2, (y_1 + y_2)/2)`

The mid-point of the diagonals of the parallelogram will coincide.

So,

Coordinate of the midpoint of AC = Coordinate of mid-point of BD

Therefore,

`((x- 2)/2 ,(3-1)/2) = ((1 +1)/2, (y + 0)/2)`

Now equate the individual terms to get the unknown value. So,

`(x - 2)/2 = 1`

x =4

Similarly,

`(y + 0)/2= 1`

y = 2

Therefore

x = 4 and y = 2