Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is constant.
Now, scale the plane $$x'=Cx$$ $$y'=Cy$$
Does this mean that in the new, primed coordinates, $L'=CL$? Or does $L$ stay exactly the same, $L'=L$?
I want to understand how fixed-length objects behave when the plane is scaled down (that is, when every graph of a function y=y(x) is scaled down).
It is confusing, because all variable-related objects (functions, curves, etc) in the plane must be scaled down; but $L$, being a constant, is not related to any variables, yet is defined in the plane. Does $L$ scale down also?
Hint
Take a segment $AB$ of length $L$ based on coordinates of $A$ and $B$. Compute the image of $A \to A^\prime$ and $B \to B^\prime$ by the scaling. What is the length of the segment $A^\prime B^\prime$?