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प्रश्न
If two vertices of a parallelogram are (3, 2) (-1, 0) and the diagonals cut at (2, -5), find the other vertices of the parallelogram.
उत्तर
We have a parallelogram ABCD in which A (3, 2) and B (-1, 0) and the co-ordinate of the intersection of diagonals is M (2,-5).
We have to find the coordinates of vertices C and D.
So let the co-ordinates be `C(x_1, y_1)` and `D(x_2, y_2)`
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,
Therefore
`((3 + x_1)/2, (2 + y_1)/2) = (2,-5)`
Now equate the individual terms to get the unknown value. So,
x = 1
y = -12
So the co-ordinate of vertex C is (1,-12)
Similarly,
Co-ordinate of mid-point of BD = Co-ordinate of M
Therefore
`((-1+ x_2)/2,(0 + y_2)/2) = (2,-5)`
Now equate the individual terms to get the unknown value. So,
x = 5
y = -10
So the co-ordinate of vertex C is (5,-10)
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