Rotation in Mathematics

1. Definition of Rotation

Rotation in mathematics refers to the transformation of a shape around a fixed point called the center of rotation. When a shape is rotated, it maintains its size and shape but changes its orientation. The rotation can be clockwise (CW) or counterclockwise (CCW).

2. Properties of Rotation

• Fixed Center of Rotation: The point around which the figure rotates remains unchanged.
• Angle of Rotation: The degree by which a figure is rotated (e.g., 90°, 180°, 270°, or 360°).
• Direction of Rotation: It can be clockwise or counterclockwise.
• Shape and Size Preservation: Rotation is a rigid transformation, meaning the figure remains congruent to the original.

3. Common Rotations in the Cartesian Plane

In coordinate geometry, rotations are performed around the origin (0,0) unless stated otherwise. The general rotation rules for rotating a point $(x, y)$ are:

i. 90° Counterclockwise (CCW) Rotation

Example: Rotating (3,2) by 90° CCW results in (-2,3).

ii. 180° Rotation (Same for CW and CCW)

Example: Rotating (3,2) by 180° results in (-3,-2).

iii. 270° Counterclockwise (or 90° Clockwise) Rotation

Example: Rotating (3,2) by 270° CCW (or 90° CW) results in (2,-3).

iv. 360° Rotation

Since 360° brings the shape back to its original position, the coordinates remain unchanged.

4. Rotational Symmetry

A figure has rotational symmetry if it looks the same after being rotated by a certain angle less than 360°. The order of rotational symmetry is the number of times the shape looks the same during a full rotation.

Examples:

• An equilateral triangle has rotational symmetry of order 3 (rotates every 120°).
• A square has rotational symmetry of order 4 (rotates every 90°).

5. Real-Life Applications of Rotation

• Clock hands move in a circular motion, demonstrating rotation.
• Windmills and fans rotate around a fixed center.
• Gears and wheels in machines function based on rotational motion.

6. Conclusion

Rotation is a fundamental concept in geometry and coordinate transformations. Understanding rotation helps in solving problems related to symmetry, transformations, and real-life applications like engineering, architecture, and physics.