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Question
If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
Solution
Suppose the coordinates of parallelogram ABCD are A(1, 2), B(4, y), C(x, 6) and D(3, 5) respectively. Then by the properties of parallelogram, diagonals AC and BD intersect at point O. Let's bifurcate.
Therefore, O is the mid-point of AC and BD.
⇒ Coordinates of mid-point of AC = Coordinates of mid-point of BD
⇒ `overlinex = (x + 1)/2 = (4 + 3)/2`
⇒ x + 1 = 7
⇒ x = 6
If O is the mid-point of BD, then the coordinates of O are
`overliney = (y + 5)/2 = (6 + 2)/2`
⇒ y + 5 = 8
⇒ y = 3
Since both the coordinates are of the same point O,
⇒ x = 6 and y = 3
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