Translations of Plane Figures | Traditional Summary
Contextualization
Translation is a geometric transformation that involves moving a flat shape from one position to another without changing its form, size, or orientation. In the Cartesian plane, this movement is described by translation vectors, which indicate the direction and magnitude of the displacement. Translation is one of the basic geometric transformations, alongside rotation, reflection, and dilation, and is essential for the study of symmetries and geometric patterns.
A practical example of translation can be observed when moving an object, such as a piece of furniture, from one place to another in a room. The object maintains its original characteristics, but its position in space is altered. Similarly, when translating a geometric figure in the Cartesian plane, we apply a translation vector to its coordinates, resulting in a new position for the figure while preserving its essential properties, such as angles and lengths.
Definition of Translation
Translation is a geometric transformation that involves moving a flat shape from one position to another. During translation, the figure maintains its original properties, such as form, size, and orientation, meaning it is not resized, rotated, or distorted. Translation can be visualized as a linear movement of the figure along a vector, which indicates the direction and distance of the displacement.
In the Cartesian plane, translation is represented by vectors. These vectors have two components: one indicating displacement in the x direction and another in the y direction. For example, a translation vector (3, 2) indicates that the figure should be moved 3 units to the right and 2 units up.
Using vectors facilitates the mathematical description of translation and allows the coordinates of the points of the original figure to be adjusted accurately. When a figure is translated, each point of the original figure is moved according to the translation vector, resulting in a new position for each point.
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Translation is the displacement of a figure in the plane without altering its form, size, or orientation.
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It is represented by vectors indicating the direction and distance of the displacement.
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Each point of the original figure is moved according to the translation vector.
Translation Vectors
Translation vectors are essential tools for describing the movement of figures in the Cartesian plane. A translation vector is defined by two components: one in the horizontal direction (x) and another in the vertical direction (y). These vectors allow us to specify precisely how and where a figure will be moved.
For example, if we want to translate a figure 5 units to the right and 3 units up, we will use the translation vector (5, 3). This vector indicates that each point of the original figure will be moved 5 units in the x direction and 3 units in the y direction. The application of the translation vector is done by adding the vector's components to the original coordinates of the figure's points.
Translation vectors are especially useful in contexts where we need to perform systematic and repetitive movements, such as creating geometric patterns and animating objects in computer graphics.
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Translation vectors have two components: one in the x direction and another in the y direction.
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They specify the direction and distance of the movement of the figure.
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The application of the vector involves adding its components to the coordinates of the figure's points.
Properties of Translation
Translation has several important properties that are preserved during the displacement of the figure. One of the main properties is the preservation of angles and lengths. This means that, after translation, the measurements of the sides and the internal angles of the figure remain unchanged.
Another important property is the maintenance of parallelism between lines. If two lines are parallel in the original figure, they will continue to be parallel after translation. Furthermore, translation preserves congruence, meaning the original figure and the translated figure are congruent since they have the same form and size.
These properties make translation a powerful tool in geometry as they allow figures to be moved accurately without distorting their fundamental characteristics. Understanding these properties is crucial for solving geometric problems and for practical applications in design and engineering.
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Translation preserves angles and lengths.
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It maintains parallelism between lines.
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The original and translated figures are congruent.
Practical Application of Translation
Translation is widely used in various fields, from education to technology. In education, translation is a fundamental tool for teaching geometry and symmetry. It helps students understand how figures can be moved in the plane without altering their essential properties.
In computer graphics and animation, translation is used to move objects smoothly and controlled. For example, when a character moves from one point to another in an animation, translation is applied to calculate the new position of the character in each frame of the animation.
In graphic design, translation is employed in the creation of repetitive patterns, such as textures and prints. By systematically translating a figure, designers can create patterns that repeat harmoniously and aesthetically pleasingly. Furthermore, translation is crucial in civil engineering and architecture, where it is used to position objects and structures accurately in plans and projects.
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Translation is fundamental in teaching geometry.
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It is used in computer graphics and animation to move objects.
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It is employed in graphic design to create repetitive patterns.
To Remember
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Translation: Displacement of a figure in the plane without altering its form, size, or orientation.
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Translation Vectors: Tools that specify the direction and distance of a figure's movement.
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Cartesian Plane: Coordinate system used to describe the position of points and figures in the plane.
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Congruence: Property of figures that have the same form and size.
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Parallelism: Property of lines that remain equidistant and never intersect.
Conclusion
Translation is a geometric transformation that involves moving a flat shape from one position to another without changing its form, size, or orientation. During the lesson, we discussed the definition of translation, the use of translation vectors, and the properties preserved during movement, such as angles and lengths. The practical application of translation was exemplified in problems in the Cartesian plane and in everyday contexts, such as animations and graphic design.
Understanding the properties of translation is essential for solving geometric problems and applying these concepts in various areas, from education to technology. The preservation of angles, lengths, and parallelism between lines makes translation a powerful tool for moving figures accurately without distorting their characteristics. By mastering these concepts, students can explore practical and creative applications in different fields.
The importance of knowledge gained about translation goes beyond the classroom. By understanding how to apply translation vectors and recognizing the properties involved, students are prepared to face geometric challenges and utilize this skill in future projects and activities. We encourage students to continue exploring the topic, practicing with different figures and translation vectors to deepen their understanding.
Study Tips
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Practice drawing and translating different geometric figures in the Cartesian plane using various translation vectors.
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Use dynamic geometry software or drawing applications to visualize and experiment with translations of figures.
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Solve additional problems and exercises available in textbooks or online resources to reinforce the concept and application of translations.
