To rotate a figure 90 degrees clockwise on a coordinate plane, you transform each point (x,y) into new coordinates (y, -x).
When you rotate a figure, you are moving every point of that figure around a fixed point, called the center of rotation, which is typically the origin (0,0) unless otherwise specified. For a 90-degree clockwise rotation, the orientation of the figure changes as if it's turning to the right by a quarter turn.
Understanding the 90° Clockwise Rotation Rule
The fundamental rule for a 90-degree clockwise rotation around the origin is straightforward:
For every point (x,y) in the figure, its new position after a 90° clockwise rotation will be (y, -x).
Let's break down what this transformation means for the coordinates:
- The original
x-coordinate moves to the position of the newy-coordinate. - The original
y-coordinate moves to the position of the newx-coordinate. - The sign of the new
y-coordinate is flipped (it becomes negative if it was positive, and positive if it was negative).
Step-by-Step Guide to Rotating a Figure 90° Clockwise
Follow these simple steps to rotate any point or figure 90 degrees clockwise around the origin:
- Identify the Coordinates: For each vertex (corner point) of your figure, note its original
(x,y)coordinates. - Apply the Rule: Take the
y-coordinate from the original point and make it the newx-coordinate. Then, take thex-coordinate from the original point and make it the newy-coordinate, remembering to change its sign (negate it). - Plot the New Points: Once you have the new
(x',y')coordinates for all vertices, plot these new points on the coordinate plane. - Connect the New Points: Connect the newly plotted points in the same order as the original vertices to form the rotated figure.
Example: Rotating a Point
Let's rotate a point P(2, 5) 90 degrees clockwise around the origin.
- Original Point:
(x, y) = (2, 5) - Apply Rule:
(y, -x) - New Point:
(5, -2)
So, the point P(2, 5) rotates to P'(5, -2).
Coordinate Transformation Summary
The following table summarizes the transformation for a 90-degree clockwise rotation:
| Original Point (x,y) | Transformed Point (y,-x) |
|---|---|
| (1, 3) | (3, -1) |
| (-4, 2) | (2, -(-4)) = (2, 4) |
| (0, 6) | (6, -0) = (6, 0) |
| (-5, -1) | (-1, -(-5)) = (-1, 5) |
Why This Rule Works
This rule is derived from trigonometry and geometry, where rotating a point around the origin involves changing its angle and preserving its distance from the origin. The transformation (x,y) → (y,-x) mathematically represents this specific quarter-turn rotation in the clockwise direction. For more details on geometric transformations, you can explore resources like Khan Academy's Geometry section.
Understanding this rule makes it easy to visualize and perform 90-degree clockwise rotations on any figure on the coordinate plane.
