Show that the points `A(1,3,4),\ B(-1,6, 10),\ C(-7,4,7)a n d\ D(-5,1,1)` are have vertices of a rhombus.

Explanation

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