Question

Let $A=(1,2,3)B=(-1,-2,-1)C=(2,3,2)$ and $D=(4,7,6)$. Then $ABCD$ is a

• A. rectangle
• B. square
• C. parallelogram
• D. none of these

Answer 1

The correct option is C parallelogram

AB$=$ $\sqrt{{(-1-1)}^2+{(-2-2)}^2+{(-1-3)}^2}$
AB$=$ $\sqrt{{(-2)}^2+{(-4)}^2+{(-4)}^2}$
AB$=$ $\sqrt{36}$
AB$=$ 6
Similarly you find that BC$=$ $\sqrt{43}$ CD$=$ 6 and DA$=$ $\sqrt{43}$
Hence opposite sides of quadrilateral are equal, Now we check the diagonals
AC$=$ $\sqrt{{(2-1)}^2+{(3-2)}^2+{(2-3)}^2}$
AC$=$ $\sqrt{3}$
similarly BD$=$ $\sqrt{155}$
Diagonals are not equal
direction ratio of line passing through AB is (-2,-4,-4)
direction ratio of line passing through CD is (2,4,4), As the dr of AB and CD are proportional which means AB is parallel to CD,
Similarly check for BC and DA then you will find that they are also parallel
hence it is parallelogram