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Question
If the points A (a, -11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.
Solution
Let ABCD be a parallelogram in which the coordinates of the vertices are A (a,-11); B (5, b); C (2, 15) and D (1, 1).
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 of AC = Coordinate of mid-point of BD
Therefore,
`((a + 2)/2, (15 - 11)/2) = ((5 + 1)/2 , (b + 1)/2)`
Now equate the individual terms to get the unknown value. So,
`(a+ 2)/2 = 3`
a = 4
Similarly,
`(b + 1)/2 = 2`
b = 2
Therefore, a = 4 and b = 3
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