If you take a sheet of paper x by y and fold it so the opposite corners meet...?

Question

...what is the length of the resulting crease in terms of x and y? I figured it out once, but can't remember how I did it. My equation, however, needed x to be the shorter side and y to be the longer.

Answer 1

It's hard to write the proof of this here, because it's a bit visual, but the answer is, for x<=y:

x sqrt(1 + (x/y)^2 )

For y<=x, the answer is y sqrt(1+(y/x)^2)

Not that if y is huge relative to x, then this is close to x. That's because the fold is very close to the non-diagonal folding in half.

When y=x, then the value is the expect sqrt(2).

Answer 2

I like this problem!

Let x be the shorter side.

Let the sheet be defined by the rectangle ABCD, and A & C are the corners which meet after folding.

The triangle ABC has the diagonal length sqrt(x^2+y^2).

The crease will be at right angles to the line AC, and so we can form a smaller similar triangle where the diagonal is the crease, the long side is the width of the paper (x) and the short side runs along part of the edge of the paper.
This triangle is reduced in size by the ratio x/y, so the length of the crease is...
(x/y)sqrt(x^2+y^2)

Viola!

Rick

Answer 3

have you tried using the Pythagorean theorem?
a^2+b^2=c2.
I tried it with a piece of paper, and it seems that the short side becomes both the a and b, of the triangle with the crease as the hypotenuse.

Answer 4

X squared + Y squared = length of crease squared