Rotating coordinates involves transforming a point's position on a plane around a central point (usually the origin) by a specific angle. The rules for rotating coordinates depend on the angle and direction of rotation.
Specific Coordinate Rotation Rules
While rotations can be done for any angle, specific degree rotations have simple rules that can be applied directly to the coordinate pair (x, y). Based on the provided information, we can outline the rules for two common rotations:
Rotating 180 Degrees Clockwise
To rotate a point 180 degrees clockwise around the origin:
- Change the sign of both the x-coordinate and the y-coordinate.
If a point has coordinates (x, y), its coordinates after a 180-degree clockwise rotation become (-x, -y).
Example:
According to the reference, for the point (-2, 3), a 180-degree clockwise rotation would give us (2, -3). The sign of -2 changed to +2, and the sign of +3 changed to -3.
Rotating 90 Degrees Counterclockwise
To rotate a point 90 degrees counterclockwise around the origin:
- Swap the coordinates.
- Change the sign of the new first coordinate (the original y-coordinate).
If a point has coordinates (x, y), its coordinates after a 90-degree counterclockwise rotation become (-y, x).
Example:
Let's apply this rule to the point (3, -5):
- Swap the coordinates: (-5, 3)
- Change the sign of the first coordinate (-5): (5, 3)
The point (3, -5) rotated 90 degrees counterclockwise becomes (5, 3).
Summary of Rules
Here's a quick reference for the specific rotations mentioned:
| Rotation | Rule | Transformation (x, y) -> (?, ?) | Example (-2, 3) -> (?, ?) |
|---|---|---|---|
| 180 Degrees Clockwise | Change signs of both coordinates | (-x, -y) | (2, -3) |
| 90 Degrees Counterclockwise | Swap coordinates, change sign of first | (-y, x) | (-3, -2) |
Note: The example for 90 degrees counterclockwise uses the point (-2, 3) for consistency with the 180-degree example. Applying the rule to (-2, 3): swap to (3, -2), change sign of first gives (-3, -2).
These rules provide specific transformations for points rotated by these exact degree amounts around the origin.
