To rotate a point \((x, y)\) clockwise 180 degrees about the origin, we can use the following transformation:
\[ (x, y) \rightarrow (-x, -y) \]
For the point \((1, 2)\):
\[ (1, 2) \rightarrow (-1, -2) \]
Thus, after rotating the point \((1, 2)\) 180 degrees clockwise about the origin, the new point is \((-1, -2)\).