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Question
Three consecutive vertices of a parallelogram are (-2,-1), (1, 0) and (4, 3). Find the fourth vertex.
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Solution
Let ABCD be a parallelogram in which the coordinates of the vertices are A (−2,−1); B (1, 0) and C (4, 3). We have to find the coordinates of the fourth vertex.
Let the fourth vertex be D(x,y)
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
Now 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,
Co-ordinate of mid-point of AC = Co- ordinate of midpoint of BD
Therefore
`((x+1)/2, y/2) = ((4-1)/2,(3-1)/2)`
`((x + 1)/2, y/2) = (1,1)`
Now equate the individual terms to get the unknown value. So,
x = 1
y = 2
So the forth vertex is D(1, 2)
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