To determine which rule describes the transformation of a triangle rotated [tex]\(90^\circ\)[/tex] about the origin, let's analyze each potential transformation rule:
1. [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]: This rule describes a 180-degree rotation about the origin since it flips both the x and y coordinates to their opposites.
2. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]: This rule describes the coordinates after a 90-degree counterclockwise rotation about the origin. Here, the original x-coordinate becomes the new y-coordinate (with a sign change), and the original y-coordinate becomes the new x-coordinate.
3. [tex]\((x, y) \rightarrow (-y, -x)\)[/tex]: This rule describes a 270-degree counterclockwise or 90-degree clockwise rotation about the origin as it flips the coordinates and changes the signs.
4. [tex]\((x, y) \rightarrow (y, -x)\)[/tex]: This rule describes a 90-degree clockwise rotation about the origin since the original x-coordinate becomes the negative y-coordinate, and the original y-coordinate becomes the x-coordinate, but with a sign change.
Given the question specifies a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation, the rule that best fits this transformation is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Therefore, the rule that describes the transformation of the triangle rotated 90 degrees counterclockwise about the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
So, the correct rule is:
[tex]\[ \boxed{2} \][/tex]
