Or with arbitrarily sized floor.
Take an even number N, larger than 2
Assume floor has either N by N, or N+1 by N+1 tiles
Drawn the line from 0,0 to N,N (let us say these are two intersections with our rectangular tiles)
mark the point (N/2,N,2), let us say this is also rectangular carpet intersection, now we have three intersections, and one side of out rectangle is N/sqrt(2)
choose the other side of the rectangle to be an transcendental number T, this number is not the root of a polynomial equation with rational coefficients. As the tile intersections are M.sqrt(2)N away perendicular to the original line and the polygon intersections are K.T away, where M, K, N are integers the there can be no other intersections otherwise T = M.sqrt(2).N/K which would mean T was not transcendental.