To determine the rule for rotating a point with coordinates [tex]\((x, y)\)[/tex] by 90 degrees counterclockwise about the origin, let's analyze each option given:
A. [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
B. [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
C. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
D. [tex]\((x, y) \rightarrow (y, x)\)[/tex]
When a point [tex]\((x, y)\)[/tex] is rotated 90 degrees counterclockwise about the origin, the new coordinates of the point can be determined by understanding the effect of this transformation on the coordinate system. Here are the detailed steps:
1. Start with the initial point [tex]\((x, y)\)[/tex].
2. After rotating 90 degrees counterclockwise around the origin, the new coordinates will effectively swap and change signs based on their positions in the coordinate plane.
3. Specifically, the original point [tex]\((x, y)\)[/tex] will become [tex]\((-y, x)\)[/tex]. This is because:
- The [tex]\(x\)[/tex]-coordinate, moving to a new position in the [tex]\(y\)[/tex]-axis, becomes [tex]\(-y\)[/tex].
- The [tex]\(y\)[/tex]-coordinate, moving to a new position in the [tex]\(x\)[/tex]-axis, becomes [tex]\(x\)[/tex].
Thus, the correct rule for rotating a point [tex]\((x, y)\)[/tex] 90 degrees counterclockwise about the origin is:
[tex]\[
(x, y) \rightarrow (-y, x)
\][/tex]
From the given options, the correct choice is:
C. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
