Three vertices of a parallelogram taken in order are (-1, 0), (3, 1) and (2, 2) respectively. Find the coordinates of the fourth vertex.

Let A(-1, 0), B(3, 1), C(2, 2) and D(x, y) be the vertices of a parallelogram ABCD taken in order.

Since, the diagonals of a parallelogram bisect each other,

∴ Coordinates of the mid-point of AC = Coordinates of the mid-point of BD

As we know the coordinates of A and C both, we find the midpoint AC.

Midpoint of AC:

x- coordinate = (x₁ + x₂) / 2

= (-1 + 2 ) / 2 = 1/2

y-coordinate = (y₁ + y₂) / 2

= (0 + 2 ) / 2 = 1

Mid point of AC = ( 1/2 , 1)

As mid point of AC = mid point of BD

Mid point of BD = ( 1/2 , 1)

x- coordinate = (x₁ + x₂) / 2

= ( x + 3 ) / 2

⇒ (x + 3)/2 = 1/2
⇒ x + 3 = 1
⇒ x = -2

y- coordinate = (y₁ + y₂)/2

= (y + 1 ) / 2

⇒ (y + 1) /2 = 1
⇒y + 1 =2
⇒y=1

∴ The coordinates of the fourth vertex, D are ( -2 , 1 ).