An example of rotation on a coordinate plane involves turning a point or shape around a fixed point, called the center of rotation, by a specific angle and in a particular direction.
Understanding Rotation
Rotation is a fundamental type of transformation in geometry. Unlike translations (slides) or reflections (flips), rotation involves pivoting an object. Key aspects of a rotation include:
- Center of Rotation: The fixed point around which the object turns. Often, this is the origin (0,0) on a coordinate plane, but it can be any point.
- Angle of Rotation: The number of degrees the object is turned. Common angles are 90°, 180°, and 270°.
- Direction of Rotation: Whether the rotation is clockwise or counterclockwise. Counterclockwise rotation is usually considered the positive direction.
A Specific Example of Rotation
Based on the provided reference, here is a clear example of a point being rotated on a coordinate plane:
Rotate the point (5, 8) about the origin 270° clockwise.
Let's break down this example:
- Original Point: The point being rotated is P(5, 8). Here, the x-coordinate is 5 and the y-coordinate is 8.
- Center of Rotation: The rotation is performed "about the origin," which is the point (0,0).
- Angle and Direction: The rotation is 270 degrees in the clockwise direction.
Applying the Rotation Rule
The reference provides a specific rule for rotating an object 270° clockwise about the origin:
- Rule: Take the opposite value of the x-coordinate and then switch it with the y-coordinate.
Let's apply this rule to our point (5, 8):
- The original x-coordinate is 5. Its opposite value is -5.
- The original y-coordinate is 8.
- According to the rule, we switch the coordinates and use the opposite of the original x-coordinate in the new y-position.
So, the new point, P', is obtained by swapping the coordinates (y, x) and negating the original x-coordinate in its new position: (y, -x).
Applying this to (5, 8):
- The new x-coordinate becomes the original y-coordinate, which is 8.
- The new y-coordinate becomes the opposite of the original x-coordinate, which is -5.
Therefore, the rotated point is (8, -5).
Here's a summary in a table:
| Aspect | Before Rotation | After Rotation |
|---|---|---|
| Original Point (x, y) | (5, 8) | |
| Rotation Parameters | Origin (0,0), 270° Clockwise | |
| Rule Applied | Swap coordinates, negate original x | |
| Rotated Point (x', y') | (8, -5) |
This example clearly demonstrates how a point's coordinates change when subjected to a specific rotation around the origin on a coordinate plane.
