如图,在平面直角坐标系中,$\triangle ABC$的顶点坐标分别为$A\left(1,0\right)$,$B\left(4,2\right)$,$C\left(2,4\right)$.
$(1)$画出$\triangle ABC$关于原点$O$对称的$\triangle A_{1}B_{1}C_{1}$；（点$A$、$B$、$C$的对应点分别为$A_{1}$、$B_{1}$、$C_{1})$
$(2)$将$\triangle ABC$绕着点$A$顺时针旋转$90^{\circ}$得到$\triangle AB_{2}C_{2}$,画出$\triangle AB_{2}C_{2}$,并写出点$B_{2}$的坐标.(点$B$、$C$的对应点分别为$B_{2}$、$C_{2})$