Verify the following:

(5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are the vertices of a rhombus.

Given: Points are A(5, -1, 1), B(7, -4, 7), C(1, -6, 10) and D(-1, -3, 4)


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


All sides of both square and rhombus are equal


But diagonals of a rhombus are not equal whereas they are equal for square


Formula used:


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



Therefore,


Distance between A(5, -1, 1) and B(7, -4, 7) is AB,






= 7


Distance between B(7, -4, 7) and C(1, -6, 10) is BC,






= 7


Distance between C(1, -6, 10) and D(-1, -3, 4) is CD,






= 7


Distance between A(5, -1, 1) and D(-1, -3, 4) is AD,






= 7


Clearly,


AB = BC = CD = AD


All sides are equal


Now, we will find length of diagonals


Distance between A(5, -1, 1) and C(1, -6, 10) is AC,






Distance between B(7, -4, 7) and D(-1, -3, 4) is BD,






Clearly,


AC BD


The diagonals are not equal, but all sides are equal


Thus, Quadrilateral formed by ABCD is a rhombus


Hence Proved


20