解: （3）如图，连接$OA，$$OA_{2}，$以点$O$为圆心、$OA$长为半径作$\overset{\frown}{AA_{2}}。$由题意，易得$OA = OA_{2}=\sqrt{4^{2}+1^{2}}=\sqrt{17}，$$\angle AOA_{2}=90^{\circ}，$$\therefore\overset{\frown}{AA_{2}}$的长为$\frac{90\pi\times\sqrt{17}}{180}=\frac{\sqrt{17}}{2}\pi，$即点$A$所经过的路径长为$\frac{\sqrt{17}}{2}\pi。$
