4. In Rt $\triangle A B C$, $\angle B C A=90^{\circ}$, $C D$ is the altitude. It is known that the three sides of $\mathrm{Rt} \triangle A B C$ are all integers, and $B D=11^{3}$. Find the ratio of the perimeters of $\mathrm{Rt} \triangle B C D$ and $\mathrm{Rt} \triangle A C D$.
(Given: Let the three sides of $\triangle A B C$ be $a, b, c$. From $11^{3} c = a^{2}$, set $a = 11^{2} p, c = 11 p^{2}$, then $b^{2} = c^{2} - a^{2} = (11 p)^{2} (p^{2} - 11^{2})$. Set $b = 11 p q$, then $p^{2} - q^{2} = 11^{2}$. Therefore, $p = 61, q = 60$. Answer: $\frac{11}{60}$.)