Show that the points A(1, 1, 1), B(-2, 4, 1), C(1, -5, 5) and D(2, 2, 5) are the vertices of a square.
To prove: Points A, B, C, D form square.
Formula:
The distance between two points (x1,y1,z1) and (x2,y2,z2) is given by
D=
Here,
(x1,y1,z1)= (1, 1, 1)
(x2,y2,z2)= (-2, 4, 1)
(x3,y3,z3)= (-1, 5, 5)
(x4,y4,z4)= (2, 2, 5)
Length AB = 
= 
= 
= 
= 
Length BC = 
= 
= 
= 
= 
Length CD = 
= 
= 
= 
= 
Length AD = 
= 
= 
= 
= 
Length AC = 
= 
= 
= 
= 
Length BD = 
= 
= 
= 
= 
Here, AB = BC = CD = AD
Also, AC = BD
This means all the sides are the same and diagonals are also equal.
Hence vertices A, B, C, D form a square.
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