To find the new coordinates of a point (x, y) when it is rotated 180 degrees clockwise about the origin, you can use the following transformation:
\[ (x', y') = (-x, -y) \]
Given the point (1, 2), we can apply the transformation:
\[ x' = -1 \] \[ y' = -2 \]
Thus, the new point after a 180-degree clockwise rotation about the origin is:
\[ (-1, -2) \]
So, the answer is:
\((-1, -2)\)