To solve this problem, we need to determine the new coordinates of the point [tex]\((1, -2)\)[/tex] after a rotation of [tex]\(-180^\circ\)[/tex] about the origin. A rotation of [tex]\(-180^\circ\)[/tex] about the origin essentially flips the point to the exact opposite position across the origin.
Here is a step-by-step explanation:
1. Understand the Rotation by [tex]\(-180^\circ\)[/tex]:
Rotating a point by [tex]\(-180^\circ\)[/tex] is equivalent to rotating the point [tex]\(180^\circ\)[/tex] in the clockwise direction, which is the same as rotating it [tex]\(180^\circ\)[/tex] in the counterclockwise direction. The effect of a [tex]\(180^\circ\)[/tex] rotation is that both the x and y coordinates of the point will change signs.
2. Applying the Rotation:
The coordinates after a [tex]\(180^\circ\)[/tex] rotation from [tex]\((x, y)\)[/tex] become [tex]\((-x, -y)\)[/tex].
3. Calculate the New Coordinates:
For the point [tex]\((1, -2)\)[/tex], after a [tex]\(-180^\circ\)[/tex] rotation, the new coordinates will be:
[tex]\[
(-1 \cdot 1, -1 \cdot -2) = (-1, 2).
\][/tex]
Hence, after a rotation of [tex]\(-180^\circ\)[/tex] about the origin, the point [tex]\((1, -2)\)[/tex] moves to [tex]\((-1, 2)\)[/tex].
So, the correct answer is [tex]\(\boxed{(-1,2)}\)[/tex].
