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प्रश्न
If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.
उत्तर
Let ABCD be a parallelogram in which the coordinates of the vertices are A (6, 1); B (8, 2); C (9, 4) and D (k, p).
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,
Co-ordinate of mid-point o AC = Co-ordinate of mid-point of BD
Therefore,
`((6 + 9)/2, (4 + 1)/2) = ((k + 8)/2","(p + 2)/2))`
Now equate the individual terms to get the unknown value. So,
`(k + 8)/2 = 15/2`
k = 7
Similarly,
`(p + 2)/2 = 5/2`
p = 3
Therefore, k = 7 and p = 3
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