Introduction

Relevance of the Theme

Translations of plane figures are the gateway to a deeper study of plane symmetries and isometries. These fundamental concepts, not only in mathematics but also in physics, engineering, and architecture, allow for the understanding of the movement of figures in the plane without altering their shape or size. Grasping the idea of 'moving' an object in the plane mathematically is essential to develop the ability to visualize and reason geometrically.

Contextualization

Situated in the broader context of Geometry, Translations of Plane Figures are a crucial topic in the 7th grade. More specifically, students are exposed to this concept after gaining familiarity and proficiency with the study of points, lines, rays, angles, figures, and polygons. In addition to establishing the foundation for subsequent topics, such as symmetries and isometries, the study of translations helps in the development of important skills, such as spatial reasoning and the ability to represent and manipulate geometric information.

Theoretical Development

Components

• Translation: Refers to the movement of an object in a specific direction, without altering its shape, size, or orientation. In the plane, a translation can be understood as the displacement of a point or figure from one place to another, maintaining a constant distance and direction.

• Translation Vector: Represents the quantity and direction of movement in a translation. Mathematically, a translation vector is defined by an ordered pair of numbers, indicating the changes in the x and y coordinates of a point during the translation.

• Image and Pre-Image: In the context of translations, an original figure or point is called a pre-image, and the figure or point resulting from the movement is called an image.

Key Terms

• Translation: Movement of displacement of a point or figure in the plane, maintaining a constant distance and direction.

• Translation Vector: Represents the quantity and direction of movement in a translation.

• Image and Pre-Image: Terms used to describe the resulting figure and the original figure, respectively, after a translation.

Examples and Cases

• Translating a Point: Suppose we have the point A(2, 3) and we want to translate it 3 units to the right and 2 units upwards. The translation vector would be (3, 2). Applying this vector to point A, we obtain the new location of the point, which is (5, 5).

• Translation of Figures: If we have a triangle with vertices A(1, 1), B(2, 3), and C(4, 2), and we want to translate it according to the translation vector (-2, 1), each vertex of the triangle will be shifted by -2 units in the x direction and 1 unit in the y direction. Thus, the new locations of the vertices will be A'(-1, 2), B'(0, 4), and C'(2, 3).

• Identification of Translations: Given a figure before and after a movement, one of the main objectives is to identify if there was a translation and, if so, determine the translation vector. To do this, simply observe if the figure maintained the same shape and size and if there was only a parallel movement. If these two conditions are met, then there was a translation and the translation vector can be determined by observing how the coordinates of the points changed.

Detailed Summary

Key Points

• Translations of Plane Figures: Translations are a type of movement in the plane that involves the displacement of a figure or point from one place to another, without altering its shape, size, or orientation. It is an essential term in mathematics that helps develop spatial understanding and geometric reasoning.

• Translation Vector: The translation vector is the mathematical tool that allows us to fully specify a translation movement in the plane. It is represented by an ordered pair (x, y), where x represents the horizontal displacement and y the vertical displacement. The translation vector is the same for all points of the figure.

• Image and Pre-Image: In a translation, the original figure is called the pre-image and the resulting figure after the movement is called the image. Both the pre-image and the image have corresponding points, with the same distances and relative directions, so that the translation can be fully described by a vector.

Conclusions

• One of the main conclusions is that, in a translation, all points of the figure move in the same way. More specifically, each original point, or pre-image, is displaced in the same direction and by the same distance to become the respective point in the image.

• Representing a translation as a translation vector reveals that, in a translation, horizontal and vertical distances are preserved. That is, the distance between any two points in the pre-image is the same as the distance between the respective two points in the image.

Suggested Exercises

1. Translation of individual points: Given the point A(2, 5), perform the translation of this point by 4 units to the right and 3 units upwards. Determine the new location of the point after the translation.

2. Identification of the translation vector: Given the square figure ABCD, identify the figure that is the image of ABCD after the translation. Also, determine the translation vector.

3. Analysis of pairs of points: Given two pairs of points, one in the pre-image and the other in the image, determine if a translation has occurred and, if so, determine the translation vector.