Show that the points P(2, 3, 5), Q(-4, 7, -7), R(-2, 1, -10) and S(4, -3, 2) are the vertices of a rectangle.
To prove: Points P, Q, R, S forms rectangle.
Formula:
The distance between two points (x1,y1,z1) and (x2,y2,z2) is given by
D=
Here,
(x1,y1,z1)= (2, 3, 5)
(x2,y2,z2)= (-4, 7, -7)
(x3,y3,z3)= (-2, 1, -10)
(x4,y4,z4)= (4, -3, 2)
Length PQ = 
= 
= 
= 
= 
Length QR = 
= 
= 
= 
= 
Length RS = 
= 
= 
= 
= 
Length PS = 
= 
= 
= 
= 
Length PR = 
= 
= 
= 
= 
Length QS = 
= 
= 
= 
= 
Here, PQ = RS which are opposite sides of polygon.
QR = PS which are opposite sides of polygon.
Also the diagonals PR = QS.
Hence, the polygon is a rectangle.
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