system
Given the following post, respond as if you're commenting on Reddit:
human
Is it possible to have a rectangle, that when divided in half vertically results in 2 rectangles with the same proportions of the bigger one?
gpt
Yes, that's possible. You can derive what the proportions of such a rectangle should be.

Lets say that we have a rectangle with sides *a* and *b* (with *a* the length of the long edge). If you divide the rectangle in two, you end up with two rectangles each with dimensions *a* / 2 and *b*.

If *a* / 2 is the long edge of these new rectangles, then they certainly don't have the same proportions of the original one. So *a* / 2 has to be the short edge of the new rectangle.

The ratio of the length of the two sides the original rectangle is *a* / *b*, for the new one *b* / (*a* / 2) = 2 *b* / *a*.

We want both original and new ratios to be equal, so:

*a* / *b* = 2 *b* / *a*

Multiplying both sides by *a* and *b* yields:

*a*^2 = 2 *b*^2

or:

*a* = sqrt(2) *b*

So when the ratio between the two sides is the square root of 2, you get the desired property that cutting the rectangle in half results in two new rectangles with the same ratio.

Note that many paper formats use this ratio. The paper sizes denoted by A and B (e.g. A4) all have the sqrt(2) ratio. That means you can cut an A4 size paper in two and obtain two A5 sized pieces of paper.