To rotate a point \((x, y)\) about the origin by 180 degrees clockwise, you can use the following transformation:
\[ (x, y) \to (-x, -y) \]
Applying this transformation to the point \((1, 2)\):
\[ (1, 2) \to (-1, -2) \]
Therefore, after rotating the point \((1, 2)\) 180 degrees clockwise about the origin, the new point is \((-1, -2)\).