Rotation of coordinates about a point involves transforming the coordinates of a point to new coordinates based on a rotation around a fixed central point. This transformation changes the position of the point in a coordinate system while keeping its distance from the central point the same.
Specific Rotation Rules About a Point
The provided reference details specific rules for rotating coordinates. These rules describe how an original point with coordinates $(x,y)$ changes to a new point with transformed coordinates after being rotated about a central point. Based on standard geometric transformations, these rules are typically applied when the central point of rotation is the origin (0,0).
Here are the transformation rules for specific angles and directions from the reference:
Rotation Rules Summary
| Rotation Angle | Direction | Original Coordinates (x,y) | New Coordinates |
|---|---|---|---|
| 90° | clockwise | (x,y) | (y, −x) |
| 90° | counterclockwise | (x,y) | (−y, x) |
| 180° | clockwise | (x,y) | (−x, −y) |
| 180° | counterclockwise | (x,y) | (−x, −y) |
| 270° | clockwise | (x,y) | (−y, x) |
| 270° | counterclockwise | (x,y) | (y, −x) |
Applying the Rotation Rules
To find the new coordinates of a point $(x,y)$ after a rotation about the origin, you simply apply the rule corresponding to the desired angle and direction from the table above.
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Example 1: If you have a point at $(2, 3)$ and you want to rotate it 90° counterclockwise about the origin, the rule is $(x,y)$ becomes $(-y, x)$.
- Here, $x = 2$ and $y = 3$.
- The new coordinates are $(-y, x) = (-(3), 2) = (-3, 2)$.
- The rotated point is at $(-3, 2)$.
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Example 2: If you rotate the same point $(2, 3)$ 180° clockwise (or counterclockwise) about the origin, the rule is $(x,y)$ becomes $(-x, -y)$.
- The new coordinates are $(-x, -y) = (-(2), -(3)) = (-2, -3)$.
- The rotated point is at $(-2, -3)$.
These rules provide a direct way to determine the new position of a point when it undergoes a specified rotation about the origin $(0,0)$.
