Introduction

Relevance of the Topic

Translations in the Cartesian Plane are one of the main geometric operations that we can apply to points, shapes, and two-dimensional objects. The ability to perform, understand, and visualize translations is crucial in various areas, including arts, computer science, and, of course, mathematics. It serves as a basis for more advanced concepts such as isometries and symmetries, and is an essential component in topics such as analytic geometry and linear algebra.

Contextualization

Located within the vast field of Geometry, the study of translations is part of the exploration of spatial relationships and geometric transformations. In previous classes, you probably studied basic concepts about the Cartesian Plane (x and y axes, quadrants) and may have encountered operations such as rotation and reflection. Translations offer an additional perspective on how points and shapes can move in the plane. Moreover, understanding translations is indispensable for the study of future topics, such as dilation and the combination of various transformations.

Theoretical Development

Components

• Translation: Translation, a type of geometric transformation, changes the position of an object without altering its shape or orientation. In the context of the Cartesian Plane, a translation involves moving all points of a shape by the same distance and in the same direction. The essence of translation lies in its independence from rotations or reflections.

• Translation Vector: To describe a translation, we use a translation vector. This vector provides "instructions" on the amount to be translated and the direction in which the translation should occur. Each component of this vector (x value and y value) represents the movement of the shape along the corresponding axis.

• Invariance of Parallelism and Distances: In translation, two important properties are maintained: the parallelism of lines and the equality of distances. Regardless of where the original shape was located, after the translation, all parallel lines in the original shape will still be parallel, and all distances between points of the original shape will be the same.

Key Terms

• Cartesian Plane: A two-dimensional coordinate system where each point has a unique numerical representation in the form of an ordered pair (x, y). The Cartesian plane is divided into four quadrants, each defined by a pair of axes (x and y).

• Point: The smallest component of the Cartesian Plane. It is represented by an ordered pair (x, y), where x is the position on the horizontal axis, and y is the position on the vertical axis.

• Two-Dimensional Shape: An object with only length and width, without height. In the context of translations, two-dimensional shapes are moved in the Cartesian plane without any change in their shape or orientation.

Examples and Cases

• Moving a Point with a Translation: Imagine we have a point A (2,4) in the Cartesian Plane. If we apply a translation with a translation vector v = (3,1), point A will be moved 3 units to the right and 1 unit up. Therefore, the new point A' will be (5,5).

• Translation of a Shape: Consider a triangle ABC, where A = (0,0), B = (2,4), and C = (4,0). Using a translation vector v = (1,3), we can move each of the points of the triangle and obtain a new triangle A'B'C', which will be parallel and equidistant from the original triangle.

• Identifying a Translation in the Cartesian Plane: Given a shape PQRST, with P = (2,2), Q = (2,4), R = (4,4), and S = (5,3). If we apply a translation and obtain a new shape P'Q'R'S'T', establishing the property that all sides are parallel to the original sides and of equal length, we can conclude that a translation occurred.

Detailed Summary

Relevant Points

• Definition and Nature of Translation: Translation is a transformation that moves all points of a shape by the same distance in a specific direction. It is crucial to understand that translation is an operation independent of rotations or reflections.

• Translation Vector: The translation vector is a vector that represents the amount and direction in which the shape will be translated. It is a useful tool for describing the translation and its properties.

• Invariance Properties of Translation: Translation preserves the parallelism of lines and the equality of distances. This is a central feature of translation and one of the reasons why it is so widely used.

• Relationship of the Concept with the Cartesian Plane: Translations in the Cartesian Plane can be described as horizontal (on the x-axis) or vertical (on the y-axis) movements, with the magnitude of the movement determined by the components of the translation vector.

Conclusions

• Importance of Translation: Translation, although a simple operation, is a powerful tool for describing and analyzing the relative position of shapes in the plane. Understanding translations allows us to understand and perform other geometric transformations.

• Invariant Properties of Translation: The invariance of the parallelism of lines and the equality of distances are qualities that distinguish translations from other geometric transformations. It is important to highlight that these properties are always maintained, regardless of the initial position of the shapes.

• Practical Applications: Translations have practical applications in fields such as computer programming, graphic design, architecture, and many others. The ability to visualize and understand translations is, therefore, a valuable skill for many professionals.

Suggested Exercises

1. Exercise 1: Given a point A (2,3) in the Cartesian Plane, perform a translation of 5 units to the left and 2 units up. What is the new position of point A after the translation?

2. Exercise 2: Draw an equilateral triangle in the Cartesian Plane, having vertices A(0,0), B(2,0) and C(1,1√3). Perform a translation of 3 units to the right and 2 units up. What is the new location of vertices A, B, and C?

3. Exercise 3: Given the quadrilateral figure PQRST, with P(1,1), Q(3,3), R(3,1) and S(4,0). If a translation is applied that moves all points 2 units to the left and 1 unit down, find the coordinates of the new quadrilateral. After the translation, is the figure still a square? Why?