To find ratio of Length and Breadth of a Rectangle Given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of the sheet, what is the ratio of length to breadth of the sheet?


$x$ is the length of the fold, and $l$ and $w$ are the respective length and width of the rectangle.
By using Pythagoras' theorem repeatedly, we can state:
$w^2 + a^2 = (l-a)^2$
$w^2 + (l-b)^2 = b^2$
$w^2 + (b-a)^2 = x^2$
From the first two equations, you can derive the values of $a$ and $b$:
$a = \frac 12(l - \frac{w^2}{l})$
$b = \frac 12(l + \frac{w^2}{l})$
Put those into the third equation and equate $x = l$ (given condition) to get:
$\frac{w^4}{l^2} + w^2 = l^2$
Now let $k = \frac lw$, the ratio between length and width.
You can now simplify to:
$k^4 - k^2 - 1 = 0$
which is a biquadratic, allowing you to find $k^2 = \frac{1 + \sqrt 5}{2}$, the golden ratio.
so the required ratio is the square root of the golden ratio.
EDIT: I solved this independently, but it really looks like a dupe of Matt's solution. Apologies for that, it was completely unintentional. But I had already done the bitmap art, so I didn't want to waste it. Hopefully, it will come in handy in visualisation.
