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प्रश्न
ABCD is a parallelogram where A(x, y), B(5, 8), C(4, 7) and D(2, −4). Find:
- co-ordinates of A.
- equation of diagonal BD.
उत्तर
In parallelogram ABCD, A(x, y), B(5, 8), C(4, 7) and D(2, −4).
The diagonals of the parallelogram bisect each other.
O is the point of intersection of AC and BD
Since O is the midpoint of BD, its coordinates will be
`((2 + 5)/2, (-4 + 8)/2)` or `(7/2, 4/2)` or `(7/2, 2)`
i. Since O is the midpoint of AC also,
`(x + 4)/2 = 7/2`
`=>` x + 4 = 7
`=>` x = 7 – 4
∴ x = 3
`(y +7)/2 = 2`
`=>` y + 7 = 4
`=>` y = 4 – 7
∴ y = –3
Thus, Coordinates of A are (3, –3)
ii. `(y - y_1)/(y_2 - y_1) = (x - x_1) /(x_2 - x_1)`
`=> y - y_1 = ((y_2 - y_1))/((x_2 - x_1)) xx (x - x_1)`
`=> y + 4 = (8 + 4)/(5 - 2) xx (x - 2)`
`=> y + 4 = 12/3 xx (x - 2)`
`=>` y + 4 = 4(x – 2)
`=>` y + 4 = 4x – 8
`=>` 4x – y = 12
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