Question

Show that the quadrilateral with verticies at P(0,2,5), Q(1,6,2), R(7,4,2), and S(6,0,5) is a parallelogram.

Answer 1

vector PQ = 1 i + 4 j - 3 k
vector SR = 1 i + 4 j - 3 k
parallel

vector QR = 6 i - 2 j + 0 k
Vector PS = 6 i - 2 j + 0 k
parallel
done

Answer 2

Your figure is embedded in three dimensional space, which makes it harder.

First consider the side lengths:
PQ: sqrt(1^2 + 4^2 + 3^2) = 5
QR: sqrt(6^2 + 2^2 + 0) = sqrt40
RS: sqrt(1^2 + 4^2 + 3^2) = 5
SP: sqrt(6^2 + 2^2 + 0) = sqrt40
Next look at the direction cosines. They are the same for the PQ and RS pair, and for the pair QR and SP.

Opposite sides are of equal length and parallel. Adjacent sides are connected. It must be a parallelogram. There must be a theorem for that.

Answer 3

LOL , if opposite sides with four sides are parallel, it is a parallelogram.