# usable rectangles

This question deals with efficient division of land into land-plots.

For the sake of this question, assume that a land-plot is usable only if it is a rectangle whose width/height ratio is between $r$ and $1/r$. If a person gets a plot that is not entirely usable, he has to find a single usable rectangle that is contained within that plot; see figure 1 (where $r$ is approximately 2).

I am interested in what happens when we divide a usable land-plot. For example, in figure 2a we divide a usable plot into two plots by cutting at the middle of the narrow side. We have two land-plots, but in each of them, only about half is still usable (for $r=2$), so there is a loss of about $1/2$ (note that each citizen gets a single land-plot, and can use a single usable rectangle in the plot). However, in figure 2b we divide the same plot by cutting at the middle of the wide side, and get two rectangles with more balanced width/height ratio, therefore there is no loss at all.

What happens if we cut the rectangle with an arbitrary straight line (figure 2c)? It seems that a loss of $1/2$ is the worst case, but I could not prove it.

Another interesting question is what happens when we do several cuts simultaneously. For example, in figure 3a we use 2 simultaneous cuts that are parallel to the sides of the rectangle, and divide the rectangle into 4 plots with the same width/height ratio; therefore there is no loss at all. But what if we divide with 2 perpendicular cuts that are not parallel to the sides of the rectangle - what is the maximum possible loss? and what is the minimum loss, if we can move the cuts relative to the rectangle, but not rotate them? (i.e. the angle of the cuts relative to the rectangle sides is given, but we can place the cuts so as to maximize the usable area in each of the cuts).