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प्रश्न
The three vertices of a parallelogram are (3, 4) (3, 8) and (9, 8). Find the fourth vertex.
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उत्तर
Let A(3,4), B(3,8) and C(9,8) be the three given vertex then the fourth vertex D(x,y)
Since ABCD is a parallelogram, the diagonals bisect each other.
Therefore the mid-point of diagonals of the parallelogram coincide.
Let p(x,y) be the mid-point of diagonals AC then,
`P(x,y) = ((3 + 9)/2, (4 + 8)/2)`
P(x,y) = (6,6)
Let Q(x,y) be the mid point of diagonal BD. then
`Q(x,y) = ((3 + x)/2, (8 + y)/2)`
Coordinates of mid-point AC = Coordinates of mid-point BD
P(x,y) = Q(x,y)
`=> (6,6) = ((3 + x)/2, (8 + y)/2)`
Now equating individual components
`=> 6 = (3 + x)/2` and `6 = (8 + 4)/2`
=> 3 + x = 12 and 8 + y = 12
=> x = 9 and y = 4
Hence, coordinates of fourth points are (9, 4)
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