4. As shown in Figure 1,

Given that the circumcircle $O$ of $\triangle \mathrm{ABC}$ has diameter $A D=6 \mathrm{~cm}, \quad \angle B=$ $\angle \mathrm{DAC}$. Find the length of $\mathrm{AC}$.
4. Solution As shown in Figure 1, connect $\mathrm{CD}$,
$$
\begin{array}{l}
\because \mathrm{AD} \text { is the diameter of } \odot \mathrm{O}, \\
\therefore \angle \mathrm{ACD}=90^{\circ} . \\
\because \angle \mathrm{B}=\angle \mathrm{D}, \angle \mathrm{B}=\angle \mathrm{DAC}, \\
\therefore \angle \mathrm{DAC}=\angle \mathrm{D}, \mathrm{CD}=\mathrm{AC} .
\end{array}
$$

Thus, $2 \mathrm{AC}^{2}=36, \mathrm{AC}=3 \sqrt{2}(\mathrm{~cm})$.