Answer :

Given: Points are A(-1, 2, 1), B(1,-2, 5), C(4, -7, 8) and D(2, -3, 4)


To prove: the quadrilateral formed by these 4 points is a parallelogram


Opposite sides of a parallelogram are equal, but diagonals are not equal


Formula used:


The distance between any two points (a, b, c) and (m, n, o) is given by,



Therefore,


Distance between A(-1, 2, 1) and B(1,-2, 5) is AB,






= 6


Distance between B(1,-2, 5) and C(4, -7, 8) is BC,






Distance between C(4, -7, 8) and D(2, -3, 4) is CD,






= 6


Distance between A(-1, 2, 1) and D(2, -3, 4) is AD,






Clearly,


AB = CD


BC = AD


Opposite sides are equal


Now, we will find the length of diagonals


Distance between A(-1, 2, 1) and C(4, -7, 8) is AC,






Distance between B(1, -2, 5) and D(2, -3, 4) is BD,






Clearly,


AC BD


The diagonals are not equal, but opposite sides are equal


Thus, Quadrilateral formed by ABCD is a parallelogram


Hence Proved


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