Answer :

Given: Points are A(1, 3, 0), B(-5, 5, 2), C(-9, -1, 2) and D(-3, -3, 0)


To prove: the quadrilateral formed by these 4 points is a parallelogram but not a rectangle


Opposite sides of both parallelogram and rectangle are equal


But diagonals of a parallelogram are not equal whereas they are equal for rectangle


Formula used:


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



Therefore,


Distance between A(1, 3, 0) and B(-5, 5, 2) is AB,







Distance between B(-5, 5, 2) and C(-9, -1, 2) is BC,







Distance between C(-9, -1, 2) and D(-3, -3, 0) is CD,







Distance between A(1, 3, 0) and D(-3, -3, 0) is AD,







Clearly,


AB = CD


BC = AD


Opposite sides are equal


Now, we will find length of diagonals


Distance between A(1, 3, 0) and C(-9, -1, 2) is AC,







Distance between B(-5, 5, 2) and D(-3, -3, 0) is BD,







Clearly,


AC BD


The diagonals are not equal, but opposite sides are equal


Thus, Quadrilateral formed by ABCD is a parallelogram but not a rectangle


Hence Proved


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