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Rotation in Mathematics

Turning Shapes on the Coordinate Plane

In geometry, a Rotation is a transformation that turns a figure around a fixed point, called the center of rotation. Think of it like a wheel spinning around an axle or the hands of a clock moving around the center.

[Image of rotation geometry definition]

Like translations and reflections, rotation is a rigid transformation (isometry), meaning the size and shape of the figure do not change—only its orientation does.

1. The Basics of Rotation

To perform a rotation, you need to know three things:

  • Center of Rotation: Usually the Origin (0,0) in school mathematics.
  • Angle of Rotation: How far to turn (e.g., 90°, 180°).
  • Direction: Clockwise (CW) or Counter-Clockwise (CCW).

Note: In advanced math, the standard direction is Counter-Clockwise. Unless told otherwise, assume rotation is CCW.

2. 90° Rotation (Counter-Clockwise)

When you rotate a point 90 degrees counter-clockwise around the origin, the point moves from one quadrant to the next. The x and y coordinates swap places, and the first coordinate changes its sign.

[Image of 90 degree rotation counter clockwise]
Rule: (x, y) → (-y, x)

Example: Rotate point P(2, 5) 90° CCW.
The coordinates swap (5, 2), and the first becomes negative.
New P' = (-5, 2).

3. 180° Rotation

A 180-degree rotation is a half-turn. It moves a point to the exact opposite side of the origin. It doesn't matter if you go clockwise or counter-clockwise; you end up in the same spot.

[Image of 180 degree rotation graph]
Rule: (x, y) → (-x, -y)

Example: Rotate point A(3, -4) 180°.
Simply change the signs of both numbers.
New A' = (-3, 4).

4. 270° Rotation (Counter-Clockwise)

A 270-degree rotation counter-clockwise is exactly the same as a 90-degree rotation clockwise. The coordinates swap places, and the second coordinate changes its sign.

Rule: (x, y) → (y, -x)

Example: Rotate point B(-1, 2) 270° CCW.
Swap to (2, -1), then change the second sign.
New B' = (2, 1).

5. Summary Table (Around Origin)

Here is a quick reference for rotating a point (x, y):

  • 90° CCW: (-y, x)
  • 180°: (-x, -y)
  • 270° CCW: (y, -x)
  • 360°: (x, y) — Full circle, back to start!

Conclusion

Understanding Rotation helps us visualize movement in space. Whether you are designing a gear system or creating animations, knowing how coordinates change as they turn around a center is a fundamental mathematical skill.