224. Folded Rectangle. Two opposite vertices of a rectangle with sides $x$ and $y$ are overlapped. Find the length of the fold line.
224. Let the rectangle $ABCD$ have sides $AB=CD=x$, and $AD=BC=y$, where $x \geqslant y$. Fold vertex $C$ onto $A$; denote the fold line as $PQ$. By symmetry, $PC=PA=CQ=AQ$,

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Problems with a Twist
![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-150.jpg?height=412&width=988&top_left_y=316&top_left_x=541)

so segment $AC$ is perpendicular to $PQ$ and bisects it at point $O$. Triangle $AOP$ is similar to triangle $ABC$, hence

$$
\frac{PO}{BC}=\frac{AO}{AB}
$$

and

$$
PQ=2PO=\frac{2(AO)(BC)}{AB}=\frac{y}{x} \sqrt{x^{2}+y^{2}}
$$

[A. Wayne, S. S. M., 64, 241 (March 1964).]