Example 1. Point $P$ inside rectangle $A B C D$ is at distances of $3, 4, 5$ from $A, B, C$ respectively. Find the length of $P D$.
(1982 Shanghai Junior High School Mathematics Competition, 1987 Zhejiang Normal University Junior High School Mathematics Basic Knowledge Competition)
Connect $A C, B D$ intersecting at point $O$, and connect $P O$.
$$
\begin{array}{l}
\because A O=O C, B O=D O, \\
\therefore P O^{2}=\frac{1}{2} P A^{2} \\
+\frac{1}{2} P C^{2}-\frac{1}{4} A C^{2}, \\
P O^{2}=\frac{1}{2} P B^{2} \\
+\frac{1}{2} P D^{2}-\frac{1}{4} B D^{2} . \\
\text { Also } A C=B D, \\
\therefore P A^{2}+P C^{2}=P B^{2}+P D^{2} .
\end{array}
$$

Thus $P D=\sqrt{3^{2}+5^{2}-4^{2}}=3 \sqrt{2}$.