Introduction

Relevance of the Theme

Rotations of plane figures are a key topic in the study of geometry, and their understanding is fundamental for a variety of concepts in mathematics and other disciplines. They allow students to visualize and understand the transformation of figures into new positions, thus developing a deep understanding of space and shapes. This helps in developing visualization skills and spatial thinking, which are important not only in mathematics but also in areas such as visual arts, physics, engineering, architecture, and design.

Contextualization

Rotations of plane figures are a natural extension of the study of geometric figures and geometric transformations. After exploring reflections and translations, rotations introduce a third type of transformation that can be applied to plane figures. This variety of transformations allows students to explore how a figure can be altered and moved in the plane, thus expanding their understanding of geometry and the properties of figures. In addition, the study of rotations also provides a strong foundation for the study of more advanced topics in geometry, such as congruence and symmetry.

Theoretical Development

Components

• Rotation Axis: It is an imaginary line along which the figure rotates. Each point on the rotation axis remains fixed after the rotation. Visualization and understanding of the rotation axis are crucial to understanding the rotation of plane figures.

• Rotation Angle: Measure of how much a figure rotates around a fixed point called the center of rotation. It is measured in degrees and reflects the direction (clockwise or counterclockwise) and the amount of rotation.

• Center of Rotation: Fixed point around which a figure rotates. All points in the plane are rotated relative to this fixed point.

• Congruent Figures: Two figures are congruent if they can be perfectly overlapped, similar to two gloves of the same hand. In a figure rotation, the original figure and the resulting figure are congruent, as they are the same figure in different positions in the plane.

Key Terms

• Rotation - Rotation of a figure is the transformation by which the original figure is rotated around a fixed point in the plane, called the center of rotation, by a fixed angle and in a fixed direction. Each point of the figure rotates around the center of rotation by the same angle and in the same direction.

• Degree - Standard unit for measuring angles. A complete circle has 360 degrees.

• Clockwise and Counterclockwise - In the coordinate system, clockwise refers to the right direction and counterclockwise refers to the left direction. In figure rotations, clockwise implies a rotation in the direction of the clock hands, while counterclockwise implies a rotation in the opposite direction of the clock hands.

Examples and Cases

• Example of Simple Rotation:

  • Original Figure: A square with an upper vertex.

  • Center of Rotation: The upper vertex of the square.

  • Rotation Angle: 90 degrees counterclockwise.

  • Rotation Axis: The line passing through the upper vertex of the square.

  • Resulting Figure after Rotation: The square now has the right vertex as the upper vertex, but its other properties (sides, angles, area) remain the same.

• Example of Combined Rotation:

  • Original Figure: A rectangle with sides measuring 3 units and 5 units.

  • Center of Rotation: The center of the rectangle.

  • Rotation Angle: 180 degrees clockwise.

  • Rotation Axis: All lines passing through the center of the rectangle.

  • Resulting Figure after Rotation: The rectangle is now a different figure, but still has equal side lengths, therefore, it is still a rectangle.

Practicing these examples will help students internalize the concepts of rotation and develop their spatial thinking and visualization skills.

Detailed Summary

Key Points

• Definition of Rotation: Rotation is a geometric transformation that rotates a figure around a fixed point called the center of rotation. All points of the figure rotate in the same direction and by the same angle.

• Components of Rotations: Rotation axis, center of rotation, and rotation angle are fundamental elements to perform and describe a rotation of a figure.

• Rotation Axis: It is an imaginary line along which the figure rotates. All points on this line, the rotation axis, remain fixed after the rotation.

• Center of Rotation: It is the fixed point around which the figure rotates. All points of the figure in the plane rotate around this point during the rotation.

• Rotation Angle: It is the measure, in degrees, of how much the figure is rotated. It determines how much each point of the figure is displaced during the rotation.

• Congruent Figures: In the rotation of a figure, the original figure and the resulting figure are congruent, as they are the same figure in different positions in the plane.

Conclusions

• Properties of Rotations: Rotations preserve the size, shape, and orientation of the figure. This makes the original figure and the resulting figure congruent.

• Importance of Visualization: Rotations are a way to help students improve their visualization skills and spatial thinking, as they allow them to manipulate and see how figures move and transform in the plane.

• Applications Beyond Mathematics: The skills developed when studying rotations have applications in various areas, from visual arts to physics, engineering, architecture, and design.

Exercises

1. Simple Rotation: Rotate a simple figure, such as a square, 90 degrees clockwise around a point in the plane. Identify the center of rotation, the rotation axis, and the rotation angle.

2. Combined Rotation: Perform a combined rotation on a figure, first 180 degrees counterclockwise around a center of rotation, and then 90 degrees counterclockwise around a second center of rotation. Describe the components of the two rotations and the total effect on the figure.

3. Identification of Transformations: Given a pair of figures, one original and another resulting from a rotation, reflect and describe how it is possible to visually identify that a rotation has occurred, compared to other geometric transformations (reflection, translation).