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.
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