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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
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