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Prove that the points (2, -1), (4, 1), (...

Prove that the points (2, -1), (4, 1), (2, 3) and (0, 1) are the vertices of a square.

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Let the points are ` A(2 ,-1), B(4, 1), C(2, 3) and D(0, 1)`
`therefore" "AB^(2)=(4-2)^(2)+(1+1)^(2)=4+4=8`
`rArr" "AB=sqrt(8)=2sqrt(2)`
` " "BC^(2)=(2-4)^(2)+( 3-1)^(2)=(-2)^(2)+(2)^(2)=4+4=8`
`rArr" "BC=sqrt(8)=2sqrt(2)`
`" "CD^(2)=(0-2)^(2)+(1-3)^(2)=(-2)^(2)+(-2)^(2) = 4 +4=8`
`rArr" "CD=sqrt(8)=2sqrt(2)`
`" "DA^(2)=(2-0)^(2)+(-1-1)^(2)=(2)^(2)+(-2)^(2)=4+4=8`
`rArr" "DA=sqrt(8)=2sqrt(2)`
`" "AC^(2)=(2-2)^(2)+(3+1)^(2)=0+16=16`
`rArr" "AC=sqrt(16)= 4`
and `" "BD^(2)=(0-4)^(2)+(1 - 1)^(2)=16+0=16`
`rArr" "BD=sqrt(16)= 4`
Now, `AB=BC=CD=DA and AC= BD`
`thereforesquareABCD` is a square.
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