Finding Coordinates After Translation A Detailed Example

In the fascinating world of geometry, transformations play a pivotal role in understanding how shapes and figures can be manipulated in space. One of the fundamental transformations is translation, which involves moving a figure without rotating or resizing it. This article delves into the concept of translation, focusing on how to determine the coordinates of a point after it has been translated according to a specific rule. We will explore a practical example involving triangle ABC and its translation, providing a step-by-step explanation to enhance your understanding of geometric transformations. Mastering these concepts is crucial for anyone delving into geometry, as they form the basis for more complex geometric operations and analyses.

In geometry, a translation is a transformation that slides a figure or a point from one location to another without changing its size, shape, or orientation. Think of it as picking up a shape and moving it to a new spot without rotating or flipping it. Translations are defined by a rule that specifies how each point in the original figure (the pre-image) moves to its new location in the transformed figure (the image). This rule is often expressed in terms of how the x and y coordinates of a point change. For instance, a translation rule might say to add a certain number to the x-coordinate and subtract another number from the y-coordinate. This movement shifts the point horizontally and vertically, respectively. Understanding translations is fundamental in geometry because it helps us analyze how figures can be repositioned in space while maintaining their essential characteristics. This concept is not only important in theoretical mathematics but also has practical applications in fields like computer graphics, engineering, and architecture, where objects need to be moved and repositioned accurately. By grasping the principles of translations, you gain a powerful tool for visualizing and manipulating geometric shapes, paving the way for more advanced topics in geometry.

The Translation Rule (x, y) → (x+2, y-8)

The translation rule (x, y) → (x+2, y-8) is a concise way of expressing how a point moves in a two-dimensional plane. This rule tells us exactly how to transform the coordinates of any point. Let's break it down: the rule starts with (x, y), which represents the original coordinates of a point—the pre-image. The arrow → indicates that we are transforming these coordinates to a new set of coordinates. The expression (x+2, y-8) specifies the new coordinates—the image—after the translation. What this means in practice is that for every point, we take its original x-coordinate and add 2 to it, and we take its original y-coordinate and subtract 8 from it. This operation effectively shifts the point 2 units to the right (because we are adding to the x-coordinate) and 8 units down (because we are subtracting from the y-coordinate). Understanding this rule is crucial because it provides a clear and direct method for finding the new position of any point after the translation. Whether you're working with a single point or an entire figure, this rule applies uniformly, ensuring that the entire figure is shifted consistently. This consistent shift is what defines a translation, maintaining the shape and size of the original figure while changing its location. By mastering the interpretation and application of translation rules like this one, you can accurately predict and calculate the results of geometric translations, a skill that is invaluable in various areas of mathematics and its applications.

Pre-image of Point B (4, -5)

The pre-image of point B, with coordinates (4, -5), is the starting point for our translation. In geometric transformations, the pre-image is the original figure or point before any transformation is applied. In our case, point B is located at the coordinates (4, -5) on the coordinate plane. This means that its x-coordinate is 4, and its y-coordinate is -5. Understanding the coordinates of the pre-image is crucial because it serves as the foundation for determining the location of the image after the translation. The translation rule will be applied to these original coordinates to find the new coordinates of point B after it has been moved. The clarity with which we understand the pre-image directly impacts our ability to accurately apply the transformation rule and find the correct image. For point B (4, -5), this means we will use the values 4 and -5 in our translation rule to calculate its new position. This step is essential in the process of geometric transformations, as it links the initial state of a point to its final position after the transformation. By carefully identifying and understanding the pre-image, we set the stage for a precise and accurate translation.

To find the coordinates of B', which is the image of point B after the translation, we apply the translation rule (x, y) → (x+2, y-8) to the coordinates of the pre-image, which are (4, -5). This process involves two simple arithmetic steps: one for the x-coordinate and one for the y-coordinate. First, we take the x-coordinate of B, which is 4, and add 2 to it, as indicated by the (x+2) part of the rule. This gives us a new x-coordinate of 4 + 2 = 6. Next, we take the y-coordinate of B, which is -5, and subtract 8 from it, as indicated by the (y-8) part of the rule. This gives us a new y-coordinate of -5 - 8 = -13. By performing these two operations, we have transformed the original coordinates (4, -5) to the new coordinates (6, -13). These new coordinates represent the location of B' after the translation. Therefore, B' is located at the point (6, -13) on the coordinate plane. This straightforward application of the translation rule demonstrates how geometric transformations can be precisely calculated, allowing us to predict the new position of points and figures after they have been moved. Understanding this process is vital for solving a wide range of geometry problems and for grasping the fundamental concepts of spatial transformations.

Applying the Translation Rule

Applying the translation rule (x, y) → (x+2, y-8) is a straightforward process once you understand its components. The rule essentially tells us how to shift each point in the plane. To apply this rule to point B, which has coordinates (4, -5), we need to perform two separate calculations: one for the x-coordinate and one for the y-coordinate. Let’s start with the x-coordinate. The rule states that we need to add 2 to the original x-coordinate. In this case, the original x-coordinate of point B is 4. So, we calculate the new x-coordinate as 4 + 2, which equals 6. This means that the translated point will have an x-coordinate of 6. Next, we move on to the y-coordinate. The rule states that we need to subtract 8 from the original y-coordinate. The original y-coordinate of point B is -5. Therefore, we calculate the new y-coordinate as -5 - 8, which equals -13. This means that the translated point will have a y-coordinate of -13. By performing these two simple calculations, we have successfully applied the translation rule to point B. The new coordinates of the translated point B', are (6, -13). This process highlights the simplicity and precision of geometric translations, where a well-defined rule can accurately shift points and figures in the plane. Mastering this application is crucial for understanding more complex geometric transformations and their applications.

Calculating the New Coordinates

Calculating the new coordinates after a translation involves simple arithmetic operations based on the translation rule. In our example, the translation rule is (x, y) → (x+2, y-8), and the pre-image point B has coordinates (4, -5). To find the new coordinates of B', we perform the operations specified in the translation rule separately for the x and y coordinates. For the x-coordinate, the rule tells us to add 2 to the original x-coordinate. So, we take the original x-coordinate of B, which is 4, and add 2 to it: 4 + 2 = 6. This means the new x-coordinate of B' is 6. For the y-coordinate, the rule tells us to subtract 8 from the original y-coordinate. We take the original y-coordinate of B, which is -5, and subtract 8 from it: -5 - 8 = -13. This means the new y-coordinate of B' is -13. By performing these two calculations, we have determined the new coordinates of B' after the translation. The x-coordinate is 6, and the y-coordinate is -13. Therefore, the new coordinates of B' are (6, -13). This process demonstrates the direct and precise nature of calculating new coordinates under a translation. The translation rule provides a clear formula for how each coordinate changes, making it easy to find the new position of a point after the transformation. Understanding this calculation is fundamental for working with translations and other geometric transformations.

After applying the translation rule (x, y) → (x+2, y-8) to the pre-image of point B, which has coordinates (4, -5), we have successfully calculated the coordinates of B'. By adding 2 to the x-coordinate and subtracting 8 from the y-coordinate, we found the new coordinates to be (6, -13). This means that B' is located at the point (6, -13) on the coordinate plane. This result is the solution to our problem and demonstrates the practical application of geometric translations. The coordinates (6, -13) provide a precise location for B' after the translation, showing how the original point B has been shifted in the plane according to the given rule. This process underscores the importance of understanding translation rules and their application in geometry. By accurately applying the translation rule, we can confidently determine the new positions of points and figures, which is a fundamental skill in geometry and related fields. The solution (6, -13) not only answers the specific question but also reinforces the broader concept of how translations work and how they can be used to transform geometric shapes.

In conclusion, understanding geometric translations is crucial for anyone studying geometry, as it forms the foundation for more complex geometric concepts. In this article, we explored the concept of translation by focusing on a specific example: translating triangle ABC according to the rule (x, y) → (x+2, y-8). We successfully determined the coordinates of B' after the translation, starting from the pre-image of point B, which had coordinates (4, -5). By applying the translation rule, we calculated the new coordinates of B' to be (6, -13). This process involved adding 2 to the original x-coordinate and subtracting 8 from the original y-coordinate, demonstrating a straightforward yet powerful method for performing translations. The ability to accurately apply translation rules and find the new coordinates of points after transformation is a fundamental skill in geometry. It allows us to predict and analyze how figures move in space while maintaining their shape and size. This understanding is not only valuable in theoretical mathematics but also has practical applications in fields such as computer graphics, engineering, and architecture. By mastering the principles of geometric translations, you gain a solid foundation for tackling more advanced topics in geometry and related disciplines. This article has provided a clear, step-by-step explanation of how to perform a translation, equipping you with the knowledge and skills to confidently solve similar problems in the future.