解:?$(1)$?如圖所示,?$△A_{1}O_{1}B_{1}$?即為所求
?$(2)$?如圖所示,?$△A_{2}O_{2}B_{2}$?即為所求
?$(3)$?由圖可知,點(diǎn)?$B$?繞點(diǎn)?$O$?旋轉(zhuǎn)到點(diǎn)?$B_{2}$?
所經(jīng)過(guò)的路徑是弧?$BB_{2}$?
在?$Rt△AOB$?中,?$OB= \sqrt {OA2+AB2}=5$?
∴路徑長(zhǎng)?$=\frac {90}{360}×2π×5=\frac 52π$?