Let $A=(0,6)$ and $B=(0,10)$. Points $A^{\prime }$ and $B^{\prime }$ are on the line $y=x$, and $\overline{A{A}^{\prime }}$ and $\overline{B{B}^{\prime }}$ intersect at $C=(3,5)$. What is the length of $\overline{A^{\prime }{B}^{\prime }}$?