Finding the Ratio of Similarity Between Triangles BCD and ABC in a Rectangle

To find the ratio of similarity between triangles $\triangle BCD$ and $\triangle ABC$, we start by noting the configuration of rectangle $ABCD$.

Since $ABCD$ is a rectangle, $\angle ABC$ and $\angle BCD$ are right angles. Thus, triangles $\triangle BCD$ and $\triangle ABC$ are right triangles.

Both triangles share the same height (side length $BC = 5$) and base (in triangle $\triangle BCD, \ DC = 9$ and in triangle $\triangle ABC, \ AB = 9$).

The important observation is that despite differing configurations, these triangles maintain a proportionate structure, both sharing the same dimensions in the rectangle. This can make both triangles similar.

Thus, the ratio of similarity between the sides of $\triangle BCD$ and $\triangle ABC$ is 1.

Therefore, the solution to the problem is 1.