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Question
Prove that the points (5 , 3) , (1 , 2), (2 , -2) and (6 ,-1) are the vertices of a square.
Solution
AB = `sqrt ((5-1)^2 + (3 -2)^2) = sqrt (16 + 1) = sqrt 17` units
BC = `sqrt ((1 - 2)^2 + (2 + 2)^2) = sqrt (1 + 16) = sqrt 17` units
CD = `sqrt ((6 - 2)^2 + (-1 + 2)^2) = sqrt (16 + 1) = sqrt 17` units
DA = `sqrt ((6 - 5)^2 + (- 1 - 3)^2) = sqrt (1 + 16) = sqrt 17` units
AC = `sqrt ((5 - 2)^2 + (3 + 2)^2) = sqrt (9 +25) = sqrt 34` units
BD = `sqrt ((6-1)^2 + (-1 -2)^2) = sqrt (25 + 9) = sqrt 34` units
∵ AB = BC = CD = DA and AC = BD
∴ ABCD is a square.
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