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Question
If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides
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Solution
We know that diagonals of a parallelogram bisect each other.
Coordinates of the midpoint of AC = coordinates of the midpoint of BD
the midpoint of AC = midpoint of BD
`=> ((4-2)/2, (b+1)/2) = ((a+1)/2 , (0 +2)/2)`
`=> (2/2, (b+1)/2) = ((a+1)/2 , 2/2)`
`=> (1, (b+1)/2) = ((a+1)/2 , 1)`
So
`1 = (a+1)/2``
2 = a + 1
`:. a = 1`
and
`(b +1)/2 = 1`
`=> b + 1 = 2`
`:. b = 1`
Therefore, the coordinates are A(–2, 1), B(1, 0), C(4, 1) and D(1, 2).
`AB = DC = sqrt((1+2)^2 + (0 - 1)^2) = sqrt(9 + 1) = sqrt(10)`
`AD = BC = sqrt((1+2)^2 + (2-1)^2) = sqrt(9 + 1) = sqrt10`
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