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Question
Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.
Solution
Let the vertices of quadrilateral ABCD be A(−1, 2, 1), B(1, –2, 5), C(4, –7, 8) and D(2, –3, 4), then
AB2 = `sqrt((1 + 1)^2 + (−2 - 2)^2 + (5 - 1)^2)`
= `sqrt(4 + 16 + 16)`
= `sqrt36`
= 6
BC2 = `sqrt((4 - 1)^2+ (-7 + 2)^2 + (8 - 5)^2)`
= `sqrt(9 + 25 + 9)`
= `sqrt43`
CD2 = `sqrt((2 - 4)^2 + (−3 + 7)^2 + (4 - 8)^2)`
= `sqrt(4 + 16 + 16)`
= `sqrt36`
= 6
AD2 = `sqrt((2 + 1)^2 + (−3 - 2)^2 + (4 - 1)^2)`
= `sqrt(9 + 25 + 9)`
= `sqrt43`
AB2 = CD2 and BC2 = AD2
AB = CD and BC = AD
Hence, the given points are the vertices of a parallelogram.
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