Translations: Advanced

Relevance of the Topic

Translations, the ability to move an object in a specified direction without altering its shape or orientation, are a fundamental component of geometry. Their relevance is not limited to the study of mathematics alone, as they are applied in many disciplines, including architecture, engineering, and computer science. Furthermore, understanding translations is essential for working with various mathematical concepts, including symmetry, congruence, and similarity.

Learning advanced translations allows students to develop a deeper understanding of geometric principles and expand their problem-solving skills. With this ability, students can visualize and describe more complex geometric transformations, providing a solid foundation for future studies in mathematics and other disciplines.

Contextualization

Translations are part of the broader content of Geometric Transformations, which is one of the main units in the 3rd year High School mathematics curriculum. This topic is generally addressed after the study of congruence and symmetry and before advanced topics such as homothety and isometries.

Translations are the simplest form of transformation that preserves the size, shape, and orientation of geometric figures. For translations, no matter how much the figure is rotated or reflected, it will always maintain the same distances and angles between corresponding points.

Throughout this section, we will expand our basic knowledge of translations to explore more complex issues, such as:

• Combined operations of translations: How to perform multiple translations in sequence and identify the result of these transformations.
• Representation of translations: How to describe a translation using vectors and coordinates.
• Invariant properties in translations: Identify and demonstrate the properties that do not change after a translation.
• Application of translations: How to use translations to solve real-life problems and other more challenging mathematical problems.

Theoretical Development

Components

• Translations: A translation is the geometric transformation that moves each point of an object by a constant distance in a specified direction. During a translation, all sizes, shapes, and angles of the object remain unchanged. The direction and distance of the movement are the same for all points of the object. Translations are called "rigid motions" because the shape of the object does not change.

• Translation Vectors: Translation vectors are used to represent a translation in the Cartesian plane or in space. A translation vector has a direction and a length corresponding to the direction and distance of the translation, respectively.

• Composition Law of Translations: The composition law of translations states that the sequence of two (or more) translations is always equivalent to a single translation, whose translation vector is the sum of the translation vectors of the individual translations, in reverse order.

Key Terms

• Image: The figure transformed after a translation.

• Pre-image: The original figure, before the translation.

• Vector: A vector is a mathematical entity that has direction and magnitude. A translation vector is a vector that describes the movement of an object during a translation.

Examples and Cases

• Combined Operation of Translations: If we have a square ABCD with vertices A(1,1), B(1,3), C(3,3), and D(3,1) and translate it 2 units to the right and 3 units up, we will have a new square A'B'C'D' with vertices A'(3,4), B'(3,6), C'(5,6), and D'(5,4). This is a combined operation of two translations: a translation of 2 units to the right and another translation of 3 units up.

• Representation of Translations: Consider a translation that moves point A(1,2) to point A'(3,5). The translation vector corresponding to this translation is the vector AA' starting at A and ending at A'. That is, the vector AA' has the same direction and magnitude as the translation. In this case, the translation vector is (3-1, 5-2) = (2,3).

• Invariant Properties in Translations: In translation, the distances between any two points do not change, and the segments of a translated figure are parallel to the corresponding ones in the original figure, besides the shape of the figure remaining unchanged. In this case, the perimeter of square ABCD is equal to the perimeter of square A'B'C'D'.

Detailed Summary

Key Points:

• Nature of Translations: The movement of a translation is parallel. This means that all points move in the same direction and the same distance. It is important to emphasize that the shape, orientation, and size of the original figure are preserved.

• Translation Vectors: The relationship between vectors and translations is critical. The translation vector has direction and magnitude exactly equal to the movement of the translation. The direction of the vector indicates the direction of the translation, and its length indicates the distance. Furthermore, it is important to note that the translation vector is invariant, meaning it does not change unless the translation is altered.

• Composition Law of Translations: The composition law of translations is an essential tool for complex operations. It allows us to sequentially combine various translations and, surprisingly, discover that the combined action can be represented by a single translation.

• Invariant Properties: In a translation, we highlight the properties that remain unchanged. This includes the orientation of the segments of the figure, the distance between any two points in the figure, as well as the overall shape of the figure.

Conclusions:

• Recognition of Translations: The ability to recognize and describe advanced translations is crucial in geometry. Translations provide a unique way to think about movements in space and are the basis for many other concepts in mathematics.

• Vectors and Coordinates: The relationship between vectors and coordinates in translations is fundamental. Understanding that a translation vector is simply the difference between the corresponding coordinates of transformed points helps solidify the connection between these two concepts.

• Applications of Translations: Translations are not just a theoretical concept. They have practical applications in many areas, including computer graphics, game design, and engineering. The ability to apply and combine translations is a valuable skill for solving complex problems.

Suggested Exercises:

1. Describe the combined operation of two translations that takes point A(3,2) to point A''(7,5), where the first translation is 2 units to the right and the second is 3 units up.

2. Determine the composition law of the translations represented by the vectors T1 = (2,4) and T2 = (3,-1).

3. Identify the invariant properties of a figure after a translation of 5 units to the right and 3 units up.