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Question
Show that the following points are the vertices of a rectangle.
A (2, -2), B(14,10), C(11,13) and D(-1,1)
Solution
The given points are A (2, -2), B(14,10), C(11,13) and D(-1,1).
`AB = sqrt((14-2)^2 +{10-(-2)}^2) = sqrt((12)^2 +(12)^2) =sqrt(144+144) = sqrt(288) =12 sqrt(2) units`
`BC = sqrt(( 11-14)^2 +(13-10)^2 ) = sqrt((-3)^2 +(3)^2) = sqrt(9+9) = sqrt(18) = 3 sqrt(2) units`
` CD = sqrt((-1-11)^2 +(1-13)^2) = sqrt((-12)^2 +(-12)^2) = sqrt(144+144) = sqrt(288) = 12 sqrt(2) units`
`AD = sqrt((-1-2)^2 +{1-(-2)}^2) = sqrt((-3)^2 +(3)^2) = sqrt(9+9) = sqrt(18) =3 sqrt(2) units`
`Thus AB =CD = 12 sqrt(2) "units and " BC =AD = 3 sqrt(2) units`
Also ,
`AC = sqrt((11-2)^2 +{ 13-(-2)}^2) = sqrt((9)^2 +(15)^2) = sqrt(81+225) = sqrt(306) = 3 sqrt(34) units `
` BD = sqrt((-1-14)^2 +(1-10)^2) = sqrt((-15)^2 +(-9)^2) = sqrt(81+225) = sqrt(306) =3 sqrt(34) units`
Also, diagonal AC = diagonal BD
Hence, the given points from a rectangle
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