A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\tfrac{x}{y}$?
$\textbf{(A) } \dfrac{12}{13} \qquad$ $ \textbf{(B) } \dfrac{35}{37} \qquad$ $ \textbf{(C) } 1 \qquad$ $ \textbf{(D) } \dfrac{37}{35} \qquad$ $ \textbf{(E) } \dfrac{13}{12}$