Verify the following: (5,-1,1),(7,-4,7), (1,-6,10) and (-1,-3,4) are the vertices of a rhombus.

Explanation

Given that the point `A(5,-1,1),\ B(7,-4,7),\ C(1,-6,10)a n d\ D(-1,-3,4)`
we have to show that `A,B,C,D`are the vertices of a rhombus.
So ,`AB=sqrt((5-7)^2+(-1+4)^2+(1-7)^2)`
`=sqrt(4+9+36)`
`=sqrt(49)`
`=7`
`BC=sqrt((-1+7)^2+(6-4)^2+(7-10)^2)`
`=sqrt(36+4+9)`
`=sqrt(49)`
`=7`
`CD=sqrt((1+1)^2+(-6+3)^2+(10-4)^2)`
`=sqrt(36+4+9)`
`=sqrt(49)`
`=7`
`DA=sqrt((-1-5)^2+(-3+1)^2+(4-1)^2)`
`=sqrt(36+4+9)`
`=sqrt(49)`
`=7`
It is clear from the above `AB=BC=CD=DA`
Hence it is a rhombus .