Are the diagonals of a square are never perpendicular?

Answer 1

The diagonals of a square are always perpendicular.

Proof: Without loss of generality, assume the square has side length 1 and one vertex is at the origin. The square ABCD is given by:

A = (0,0) , B = (1,0) , C = (1,1) , D = (0,1)

The diagonals are d1=AC and d2=BD. Finding equations for each of them yields

d1 = x

d2 = 1-x (you can double check this)

So, the relative slopes are 1 and -1. Since their product is -1, they are perpendicular.