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प्रश्न
If A (1, 2) B (4, 3) and C (6, 6) are the three vertices of a parallelogram ABCD, find the coordinates of fourth vertex D.
उत्तर
Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (1, 2);
B (4, 3) and C (6, 6). We have to find the co-ordinates of the forth vertex.
Let the forth 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_2) " 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 mid -point of BD
Therefore,
`((1+6)/2 , (2+6)/2) = ((x + 4)/2 , ( y + 3)/2)`
`((x+4)/2 , (y + 3)/2 ) = (7/2, 4)`
Now equate the individual terms to get the unknown value. So,
`(x+4)/2 = 7/2`
x = 3
Similarly,
`(y + 3)/2 = 4`
y = 5
So the forth vertex is D ( 3 , 5) .
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