Question

Show that the co planar points A(2,3,2), B(4,7,6), C(1,2,3), D(-1,-2,-1) form a parallelogram

Answer 1

Require to show that the given co-planar points A(2,3,2), B(4,7,6), C(1,2,3), and D(-1,-2,-1)form a parallelogram.

Find the lengths of the sides AB, BC, CD and AD of the quadrilateral $ABCD$

Now

$AB=\sqrt{(4-2)^2+(7-3)^2+(6-2)^2}=\sqrt{4+16+16}=6$

$BC=\sqrt{(1-4)^2+(2-7)^2+(3-6)^2}=\sqrt{9+25+9}=\sqrt{43}$

$CD=\sqrt{(-1-1)^2+(-2-2)^2+(-1-3)^2}=\sqrt{4+16+16}=6$

$AD=\sqrt{(-1-2)^2+(-2-3)^2+(-1-2)^2}=\sqrt{9+25+9}=\sqrt{43}$

Observe that $AB=CD$ and $BC=AD$

That is opposite sides of the quadrilateral $ABCD$ are equal.

That is the quadrilateral $ABCD$ is a parallelogram.

Therefore, the given co-planar points form a parallelogram.