The points \(A(2,-1,4), B(1,0,-1), C(1,2,3)\) and \(D(2,1,8)\) form a
The points \(A(2,-1,4), B(1,0,-1), C(1,2,3)\) and \(D(2,1,8)\) form a
rectangle
square
rhombus
parallelogram
Solution
Given points are \(A(2,-1,4), B(1,0,-1), C(1,2,3)\) and \(D(2,1,8)\).
\(\therefore \quad A B=\sqrt{1^2+1^2+5^2}=\sqrt{27}\)
\(\begin{aligned}
& B C=\sqrt{0+2^2+4^2}=\sqrt{20} \\
& C D=\sqrt{1^2+1^2+5^2}=\sqrt{27}
\end{aligned}\)
and
\(D A=\sqrt{0+2^2+4^2}=\sqrt{20}\)
Here,
\(\begin{aligned}
& A B=C D \text { and } B C=D A \\
& A C=\sqrt{1^2+3^2+1^2}=\sqrt{11} \\
& \text{and } D B=\sqrt{1^2+1^2+9^2}=\sqrt{83} \\
& \therefore \quad A C \neq D B
\end{aligned}\)
\(\therefore A, B, C\) and \(D\) form a parallelogram.