If Hexagon DEFGHI Is Translated 8 Units Down And 3 Units To The Right, And The Coordinates Of Point F Are (-9,2), What Are The Coordinates Of F'?

In the fascinating world of geometry, transformations play a crucial role in understanding how shapes and figures can be manipulated in space. Among these transformations, translations hold a special place due to their simplicity and fundamental nature. A translation, in essence, is a rigid transformation that moves every point of a figure the same distance in the same direction. This means that the figure's size, shape, and orientation remain unchanged; it simply slides from one location to another. Imagine sliding a book across a table – that's a real-world example of a translation. In the context of coordinate geometry, translations are often described using vectors or coordinate rules, which specify the horizontal and vertical shifts applied to each point of the figure. Understanding translations is not only essential for grasping basic geometric concepts but also for more advanced topics such as vector algebra and computer graphics. This article delves into the specifics of determining the coordinates of a point after a translation, using a practical example involving a hexagon and its translated image.

Hexagon Translation Problem

At the heart of our exploration lies a problem involving the translation of a hexagon in the coordinate plane. Consider a hexagon named DEFGHI. This hexagon undergoes a translation, a geometric transformation that shifts the figure without rotating or resizing it. In this particular case, the hexagon is translated 8 units downwards and 3 units to the right. This means every point on the hexagon, including its vertices, moves 8 units in the negative y-direction and 3 units in the positive x-direction. We are given that the pre-image of point F, which we'll denote as F, has coordinates (-9, 2). The pre-image refers to the original position of the point before the translation. The question we aim to answer is: What are the coordinates of F', the image of point F after the translation? This problem exemplifies a fundamental concept in coordinate geometry – how translations affect the coordinates of points. By solving this, we gain a deeper understanding of how to apply translation rules and visualize transformations in the coordinate plane. The solution involves applying the translation rule to the coordinates of the original point, which will be detailed in the subsequent sections.

Understanding Coordinate Translations

To effectively solve the hexagon translation problem, it's essential to first grasp the concept of coordinate translations. In the coordinate plane, a translation can be described as a shift of points along the horizontal (x-axis) and vertical (y-axis) directions. This shift is typically represented by a translation vector or a coordinate rule. A translation vector, denoted as (a, b), indicates that each point is moved 'a' units horizontally and 'b' units vertically. If 'a' is positive, the point moves to the right; if negative, it moves to the left. Similarly, if 'b' is positive, the point moves upwards; if negative, it moves downwards. Alternatively, a coordinate rule expresses the translation as a transformation of coordinates. For instance, the rule (x, y) → (x + a, y + b) signifies that a point with original coordinates (x, y) is translated to a new position with coordinates (x + a, y + b). The values 'a' and 'b' correspond to the horizontal and vertical shifts, respectively, just like in the translation vector. Understanding these representations is crucial for performing translations accurately. In our hexagon problem, we are given a translation of 8 units down and 3 units to the right. This can be represented both as a translation vector and a coordinate rule, which we will use to find the new coordinates of point F.

Applying the Translation Rule

Now that we have a solid understanding of coordinate translations, let's apply this knowledge to our specific problem. We are given that hexagon DEFGHI is translated 8 units down and 3 units to the right. This translation can be represented as a coordinate rule: (x, y) → (x + 3, y - 8). This rule tells us that for any point (x, y) on the hexagon, its image after the translation will have coordinates (x + 3, y - 8). The '+3' indicates a shift of 3 units to the right along the x-axis, and the '-8' indicates a shift of 8 units down along the y-axis. We are also given that the coordinates of the pre-image of point F are (-9, 2). This means that before the translation, point F was located at the position where x = -9 and y = 2. To find the coordinates of F', the image of F after the translation, we simply apply the translation rule to these coordinates. We substitute x = -9 and y = 2 into the rule (x + 3, y - 8) to obtain the new coordinates. This process is a direct application of the concept of coordinate translations, allowing us to precisely determine the new location of point F after the transformation.

Calculating the Coordinates of F'

To calculate the coordinates of F', we take the coordinates of the pre-image of F, which are (-9, 2), and apply the translation rule (x, y) → (x + 3, y - 8). This means we add 3 to the x-coordinate and subtract 8 from the y-coordinate. So, the x-coordinate of F' will be -9 + 3 = -6, and the y-coordinate of F' will be 2 - 8 = -6. Therefore, the coordinates of F' are (-6, -6). This calculation demonstrates the practical application of the translation rule. By understanding how translations affect coordinates, we can accurately predict the new position of any point after a translation. This skill is crucial in various fields, including computer graphics, where translations are used to move objects on the screen, and in engineering, where they are used in the design and analysis of structures. The result we obtained, (-6, -6), corresponds to one of the answer choices provided in the problem, which confirms our solution.

Selecting the Correct Answer

After performing the calculation, we found that the coordinates of F' are (-6, -6). Now, we need to match this result with the answer choices provided in the problem. The answer choices are:

A. (-17, 5) B. (-6, -6) C. (-17, -1) D. (-12, -6)

By comparing our calculated coordinates (-6, -6) with the answer choices, we can clearly see that it matches option B. Therefore, the correct answer is B. (-6, -6). This step is crucial in problem-solving, as it ensures that we not only perform the calculations correctly but also select the appropriate answer from the given options. In this case, the correct answer indicates that after the hexagon DEFGHI is translated 8 units down and 3 units to the right, the image of point F, denoted as F', will be located at the coordinates (-6, -6). This concludes the solution to the problem, demonstrating a clear understanding of coordinate translations and their application in geometry.

Conclusion on Geometric Translation

In conclusion, this article has provided a comprehensive exploration of geometric translations, focusing on the practical application of finding the coordinates of a point after a translation. We began by introducing the concept of translations as rigid transformations that shift figures without altering their size, shape, or orientation. We then delved into a specific problem involving the translation of hexagon DEFGHI, where the hexagon was translated 8 units down and 3 units to the right. Given the pre-image coordinates of point F as (-9, 2), we set out to find the coordinates of its image, F', after the translation. To solve this problem, we emphasized the importance of understanding coordinate translations. This involves grasping how translations can be represented using translation vectors and coordinate rules. We learned that a translation rule, such as (x, y) → (x + a, y + b), describes how the coordinates of a point change after the translation. By applying the given translation rule (x, y) → (x + 3, y - 8) to the coordinates of F (-9, 2), we calculated the coordinates of F' to be (-6, -6). Finally, we matched this result with the answer choices and confirmed that option B, (-6, -6), was the correct answer. This exercise underscores the significance of coordinate geometry in solving translation problems. Understanding these concepts not only aids in solving mathematical problems but also provides a foundation for more advanced topics in geometry and related fields. The ability to accurately perform and visualize translations is a valuable skill in various applications, from computer graphics to engineering design.

Keywords: Geometric translations, coordinate translations, hexagon translation, pre-image coordinates, image coordinates, translation rule.