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