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.

Options

A. rhombus
B. rectangle
C. parallelogram
D. square

Answer

A

Explanation

Let (1,3,4),(-1,6,10,(-7,4,7) and (-5,1,1) be the coordinates of points A, B, C and D respectively.
`:. AB=sqrt((-1-1)^2+(6-3)^2+(10-4)^2)`
`=sqrt(4+9+36)=7`
`BC=sqrt((-7+1)^2+(4-6)^2+(7-10)^2)`
`=sqrt(36+4+9)=7`
`CD=sqrt((-5+7)^2=(1-4)^2+(1-7)^2`
`=sqrt(4+9+36)=7`
`DA=sqrt((1+5)^2+(3-1)^2+(4-1)^2)`
`=sqrt36+4+9)=7`
`AC=sqrt((-7-1)^2+(4-3)^2+(7-4)^2)`
`=sqrt(64+1+9)=sqrt74`
`and BD=sqrt((-5+1)^2+(1-6)^2+(1-10)^2)`
`=sqrt(16+25+81)=sqrt122`
`:' AB=BD=CD=DA`
But `BD ne AC`
`:. " Point A, B, C and D are the vertices of a rhombus."`