A square with side length $x$ is inscribed in a right triangle with legs of length $8$ and $15$ 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 legs of length $8$ and $15$ so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?

For the first square, refer to the diagram below:

For the second square, refer to the diagram below: