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?

Let the length be $l$, and the width be $w$. When the sheet is folded so that the diagonals meet, the fold crosses the sheet from one of the long sides to the other, and meets the long sides the same distance from the opposite corners. Let the distance from the intersection of the fold and the long side to the nearest corner be $d$, and the length of the fold be $f$.
Then, from the geometry of the fold, we can write:
$$w^2+(l-2d)^2=f^2$$
$$w^2+d^2=(l-d)^2$$
Solving for $d$ from the second equation results in:
$$d=\frac {l^2-w^2}{2l}$$
Substituting it into the second equation, and using your stipulation that $f=l$:
$$w^2-2(l^2-w^2)+\frac{(l^2-w^2)^2}{l^2}=0$$
Finally if we define the ratio $r = \frac l w$ and rearrange the above equation, we arrive at:
$$r^4-r^2-1=0$$
Thus $r^2$ is the golden ratio, so $r$ is the square root of the golden ratio.
Interesting problem!
