Prove that the point `A(1,3,0),\ B(-5,5,2),\ C(-9,-1,2)\ a n d\ D(-3,-3,0)` taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.

Explanation

Given that the point `A(1,3,0),\ B(-5,5,2),\ C(-9,-1,2)\ a n d\ D(-3,-3,0)`
we have to show that `A,B,C,D`are the vertices of a parallelogram and also ABCD is not a rectangle .
So ,`AB=sqrt((1+5)^2+(3-5)^2+(0-2)^2)`
`=sqrt(36+4+4)`
`=sqrt(44)`
`CD=sqrt((-9+3)^2+(-1+3)^2+(2-0)^2)`
`=sqrt(36+4+4)`
`=sqrt(44)`
`BC=sqrt((-5+9)^2+(5+1)^2+(2-2)^2)`
`=sqrt(36+16)`
`=sqrt(52)`
`DA=sqrt((-3-1)^2+(-3-3)^2+(0-0)^2)`
`=sqrt(36+16)`
`=sqrt(52)`
It is clear from the above `AB=CD and BC=DA`
Hence it is a parallelogram.
Again we have to show It is not a rectangle.
`AC=sqrt((1+9)^2+(3+1)^2+(0-2)^2)`
`=sqrt(100+16+4)`
`=sqrt(120)`
`BD=sqrt((-5+3)^2+(5+3)^2+(2-0)^2)`
`=sqrt(4+4+64)`
`=sqrt(72)`
So, `ACneBD`
Therefore ABCD is not a rectangle.