Introduction

Relevance of the Topic

Polygon Transformations are like the magic wands of Mathematics. They offer powers to move, rotate, and resize figures, helping us to better visualize abstract concepts and solve complex problems. Mastering these transformations is vital for understanding more advanced mathematical topics, such as symmetry, congruence, and similarity of polygons.

Contextualization

In this vast mathematical terrain, polygon transformations are situated at the heart of geometry and algebra, acting as a bridge between these two areas. They introduce fundamental terms and concepts, such as translation, rotation, reflection, dilation, and similarity, which are essential for understanding complex numbers, matrices, vectors, among others. Moreover, polygon transformations are the foundation for the construction of many visual structures that surround us, from artworks to architectural designs and engineering in general.

Theoretical Development

Components

• Translation: Translation is a transformation that "slides" a figure along the plane, without changing its size, shape, or orientation. That is, the translated figure is a mere copy of the original figure, but in a new position.

  • Translation Vector: A translation vector is a vector that defines the direction and distance that a figure is translated. Each point in the figure is moved in this direction by a distance equal to the length of the vector.

• Rotation: Rotation is a transformation that rotates a figure around a fixed point, called the center of rotation. All other points of the figure move around this center of rotation, and the distance of each point to the center of rotation does not change.

  • Rotation Angle: The rotation angle is the measure of how far the figure is rotated. It is always measured in degrees, and can be positive (clockwise) or negative (counterclockwise).

• Reflection: Reflection is a transformation that "mirrors" a figure along a line, called the axis of reflection. In reflection, each point of the original figure is "reflected" over the axis of reflection, to form the reflected figure.

  • Axis of Reflection: It is the imaginary line along which the reflection occurs. Each original point and its counterpart in the reflected figure are equidistant from the axis of reflection.

• Dilation: Dilation is a transformation that changes the size of a figure. The points in the original figure move away or towards the center of dilation, depending on the dilation factor.

  • Dilation Factor: The dilation factor is the ratio between the length of any segment in the dilated image and the corresponding length in the original object. If the dilation factor is greater than 1, the figure will enlarge; if it is less than 1, the figure will reduce.

• Similarity: Two figures are considered similar if one is the dilated, rotated, or reflected version of the other.

Key Terms

• Polygon: A two-dimensional flat closed figure, formed by line segments that do not intersect, each segment being called a side.

• Congruence: It is said that two figures are congruent if they have the same sizes and shapes.

• Image: The image of a figure after a transformation is the new figure resulting from the transformation.

• Original Object: The original object is the figure before the transformation.

• Translated / Rotated / Reflected / Dilated Object: It is the figure resulting after the translation, rotation, reflection, or dilation of the original object, respectively.

Examples and Cases

• Translation: Imagine a triangle with various points. Apply a translation to the right by 2 units. Each point of the triangle will be moved 2 units to the right and the resulting figure will be a triangle identical to the original, but shifted to the right.

• Rotation: Take a square. Rotating it 90 degrees around its center, each vertex will be displaced in the direction of the rotation, forming a new square.

• Reflection: Given a pentagon and a determined reflection axis (for example, a vertical line), reflect the pentagon along that axis. Each point in the original pentagon will be "reflected" over the vertical line of reflection, generating a reflected pentagon.

• Dilation: If we have a triangle and apply a dilation by a factor of 2 in relation to a center of dilation, each vertex of the triangle will move away from the center of dilation, increasing the distance between the vertices and the center of dilation. The result will be a larger triangle.

Detailed Summary

Relevant Points

• Importance of Polygon Transformations: Performing Polygon Transformations in the Cartesian plane is fundamental for the study of Mathematics, as it provides the necessary tools to understand complex concepts, such as congruence, similarity, and symmetry.

• Each Transformation as a Distinct Tool: Each transformation (Translation, Rotation, Reflection, Dilation) provides a unique tool, a "power" to modify the original configuration of a polygon. Each of them has specific rules that shape the transformation, and knowledge about these rules is essential to master the transformations themselves.

• Similarity as a Comprehensive Concept: The concept of similarity in Mathematics is vast and powerful. Understanding that two figures can be considered similar even after undergoing transformations (dilation, rotation, reflection) is a key to many more advanced mathematical applications.

Conclusions

• Points involved in a transformation: In the different transformations of polygons, some points are common: there is always an original object, an image after the transformation, and a specific element (vector, angle, axis of reflection, center of dilation) that defines and guides the transformation.

• Transformations Ensure Properties: It is important to note that, regardless of the transformation applied, certain properties of the original object are preserved. Numbers of sides, angles, and lengths of sides remain unchanged.

Exercises

1. Adapting Knowledge: Ask students to apply the four transformations to a square on a sheet of paper. Ask what has changed and what has remained the same?

2. Identifying Transformations: Show students a series of figures, some transformed and others not. Ask them to identify the figures that have been transformed and, if possible, what type of transformation was performed.

3. Creating Transformations: Give students a series of figures and ask them to choose a transformation - translation, rotation, reflection, or dilation - and apply it to each figure. Ask what they observe about the new resulting figure.