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Question

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

Solution
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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 midpoint of AC will be the same as coordinates of the midpoint of BD.

If (x1,y1) and (x1,y1) are two points then their midpoint is defined as (x1+x22,y1+y22).

By using the above conclusion,
(1+22,0+22)=(3+x2,1+y2)
(12,1)=(3+x2,y+12)
3+x2=12 and y+12=1
x=2 and y=1

Hence the fourth, vertex of the parallelogram is (2,1).

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