Question

For points $A(1,-1,1),B(1,3,1),C(4,3,1)$ and $D(4,-1,1)$ taken in order are the vertices of

• A. a parallelogram which is neither a square nor a rhombus
• B. rhombus
• C. as isosceles trapezium
• D. a cyclic quadrilateral

Answer 1

The correct option is A a parallelogram which is neither a square nor a rhombus

$\frac{\overline{A}+\overline{C}}{2}=\frac{\overline{B}+\overline{D}}{2}\Rightarrow$ it should be $\parallel gm$ May be square Rectangle or Rhombus

$AB=\sqrt{(1-1{)}^2+(3+1{)}^2+(1-1{)}^2}=4$

$CD=\sqrt{(4-4{)}^2+(3+1{)}^2+(1-1{)}^2}=4$

$AD=\sqrt{(4-1{)}^2+(-1+1{)}^2+(1-1{)}^2}=3$

$BC=\sqrt{(4-1{)}^2+(3-3{)}^2+(1-1{)}^2}=3$

As All sides are not equal it can be Rectangle or $\parallel elgm$ not but cant be square and Rhombus But $\overline {AB}.\overline {AD}\neq 0$ hence it should be Parallelogram