Expert: Ahmed Salami - 12/17/2008

Question
Give a vector proof that the midpoints of the sides of a square are the vertices of a square.

If you could help at all, it would be greatly appreciated.

Thanks!

Answer
Hi Lynn,
I hope you're familiar with position vectors.
Let the square have a length k and two of its sides lie on the x and y axes. The unit vectors are i and j in the x and y directions respectively.
The position vectors of the midpoints of the sides are then
(k/2)i, (k/2)j, (k/2)i + kj, ki + (k/2)j
You would find that the distances between any pair of adjacent points, which is the magnitude of their position vector difference, is the same and equal to
sqrt[(k/2)^2 + (k/2)^2]
= sqrt(k^2/2)
= k/sqr2

Regards