Example 4 As shown in Figure 4, in quadrilateral $ABCD$, $AB // CD, AD=$ $DC=DB=p, BC=q$. Find the length of diagonal $AC$.
(Mathematical Olympiad Junior Training Questions, Issue 6, 1996)
Analysis: From “$A D=D C=D B=p$”, we know that $A$, $B$, and $C$ are on the circle $\odot D$ with radius $p$. Using the properties of the circle, we can find the relationship between $A C$ and $p$, $q$.

Solution: Extend $C D$ to intersect the circle $\odot D$ with radius $p$ at point $E$, and connect $A E$. Clearly, $A$, $B$, and $C$ are on $\odot D$.
$\because A B / / C D$,
$\therefore \overparen{B C}=\overparen{A E}$.
Thus, $B C=A E=q$.
$A E=q$, so
$$
A C=\sqrt{C E^{2}-A E^{2}}=\sqrt{4 p^{2}-q^{2}} .
$$
2.2 Associating with the properties of the diameter to construct an auxiliary circle