Show that the points P(1, 3, 4), Q(-1, 6, 10), R(-7, 4, 7) and S(-5, 1, 1) are the vertices of a rhombus.

To prove: Points P, Q, R, S forms rhombus.

Formula:

The distance between two points (x₁,y₁,z₁) and (x₂,y₂,z₂) is given by

D=

Here,

(x₁,y₁,z₁)= (1, 3, 4)

(x₂,y₂,z₂)= (-1, 6, 10)

(x₃,y₃,z₃)= (-7, 4, 7)

(x₄,y₄,z₄)= (-5, 1, 1)

Length PQ =

=

=

=

=

Length QR =

=

=

=

=

Length RS =

=

=

=

=

Length PS =

=

=

=

=

Length PR =

=

=

=

=

Length QS =

=

=

=

=

Here, PQ = RS =QR = PS .

Also the diagonals PR ≠ QS.

Hence, the polygon is a rhombus as all sides are equal and diagonals are not equal.