Rectangle ABCD Translation Problem Finding Coordinates Of B'

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This article delves into a geometry problem involving the translation of a rectangle in the coordinate plane. We will explore the concepts of translation, coordinate geometry, and how to apply them to find the new coordinates of a point after a translation. This is a common topic in high school mathematics, particularly in geometry and algebra courses. Understanding these concepts is crucial for success in more advanced mathematical topics. The problem we'll be tackling involves rectangle ABCD with given vertex coordinates. The rectangle is then translated, and our task is to determine the new coordinates of one of its vertices, specifically point B'. Let's break down the problem and walk through the solution step by step. Coordinate geometry forms the bedrock of this problem. It allows us to represent geometric shapes and transformations using numerical coordinates. A translation is a rigid transformation, meaning it moves a figure without changing its size or shape. We'll apply this understanding to solve the problem at hand. The ability to visualize geometric transformations in the coordinate plane is a valuable skill. It helps in solving problems related to geometry, linear algebra, and even computer graphics. Let's embark on this mathematical journey and unravel the solution to this intriguing problem. This problem provides a practical application of coordinate geometry principles. By understanding translations and how they affect coordinates, we can solve various geometric problems and gain a deeper appreciation for the connection between algebra and geometry. The concepts covered in this problem are fundamental to many areas of mathematics and have real-world applications in fields such as engineering, computer science, and architecture. Now, let's dive into the problem itself and explore the steps required to find the coordinates of B'.

Problem Statement

Rectangle ABCD has vertex coordinates A(1,-2), B(4,-2), C(4,-4), and D(1, -4). It is translated 1 unit to the left and 3 units up. What are the coordinates of B'?

Understanding Translations in the Coordinate Plane

Before we dive into the solution, let's clarify the concept of translation in the coordinate plane. A translation is a transformation that shifts every point of a figure the same distance in the same direction. It's like sliding the figure without rotating or reflecting it. In the coordinate plane, a translation can be described by how much the figure is shifted horizontally (left or right) and vertically (up or down). These shifts are represented by changes in the x-coordinates and y-coordinates, respectively. A translation of 1 unit to the left means we subtract 1 from the x-coordinate of each point. Conversely, a translation of 1 unit to the right means we add 1 to the x-coordinate. Similarly, a translation of 3 units up means we add 3 to the y-coordinate of each point, while a translation of 3 units down means we subtract 3 from the y-coordinate. These simple rules form the basis for performing translations in the coordinate plane. Understanding these rules is crucial for solving problems involving translations, such as the one we're tackling in this article. The ability to visualize translations and their effect on coordinates is a valuable skill in geometry and related fields. It allows us to solve problems involving geometric transformations and understand how figures move in space. Now, let's apply this understanding to the specific problem at hand and find the new coordinates of point B after the translation. This concept of translation is fundamental in geometry and has numerous applications in various fields, including computer graphics, robotics, and engineering. A solid grasp of translations will help you tackle a wide range of mathematical problems and appreciate the beauty of geometric transformations. Let's move on to the next section and apply this knowledge to solve our problem.

Applying the Translation to Point B

Our problem states that rectangle ABCD is translated 1 unit to the left and 3 units up. We are asked to find the coordinates of B', which is the image of point B after the translation. We know that point B has coordinates (4, -2). To apply the translation, we need to adjust the x-coordinate and y-coordinate of B according to the given instructions. A translation of 1 unit to the left means we subtract 1 from the x-coordinate. So, the new x-coordinate of B' will be 4 - 1 = 3. A translation of 3 units up means we add 3 to the y-coordinate. Therefore, the new y-coordinate of B' will be -2 + 3 = 1. Combining these results, we find that the coordinates of B' are (3, 1). This simple calculation demonstrates how translations affect the coordinates of a point in the plane. By understanding the rules of translation, we can easily determine the new position of any point after a translation. This process is fundamental in geometry and has applications in various fields, such as computer graphics and animation. Let's reiterate the steps we took: we identified the original coordinates of point B, we understood the direction and magnitude of the translation, and we applied the translation to the coordinates of B to find the coordinates of B'. This clear and methodical approach is essential for solving geometry problems accurately. Now that we have found the coordinates of B', let's consider the answer choices provided and select the correct one. The ability to perform translations and understand their effects on coordinates is a valuable skill in mathematics and beyond. It allows us to analyze geometric transformations and solve problems involving movement and position. In the next section, we'll examine the answer choices and confirm our solution.

Identifying the Correct Answer

Having calculated the coordinates of B' to be (3, 1), we now need to compare our result with the provided answer choices. The answer choices are:

A. (7, -3) B. (1, -1) C. (3, 1) D. (5, -5)

By direct comparison, we can see that our calculated coordinates (3, 1) match answer choice C. Therefore, the correct answer is C. (3, 1). This step is crucial in problem-solving: always double-check your answer against the options provided, ensuring you've chosen the one that matches your calculations. It also serves as a final validation of your work, reducing the possibility of errors. Sometimes, students might make a correct calculation but mistakenly select the wrong answer choice due to carelessness or misreading. Therefore, taking this extra step can significantly improve accuracy. In this case, the answer choices are distinct enough that there's little room for confusion, but in some problems, the options might be very similar, requiring careful attention to detail. Now that we have confidently identified the correct answer, let's take a moment to reflect on the problem-solving process and the key concepts involved. We successfully applied the concept of translation in the coordinate plane to find the new coordinates of a point. This problem demonstrates the importance of understanding geometric transformations and their effect on coordinates. The ability to perform translations and other geometric transformations is fundamental in many areas of mathematics and has practical applications in various fields, including computer graphics, engineering, and physics. In the concluding section, we'll summarize the solution and highlight the key takeaways from this problem.

Summary and Key Takeaways

In this problem, we were given the coordinates of the vertices of rectangle ABCD and asked to find the coordinates of B' after a translation of 1 unit to the left and 3 units up. We successfully solved this problem by understanding the concept of translation in the coordinate plane. The key steps involved were:

  1. Understanding the concept of translation: A translation shifts every point of a figure the same distance in the same direction.
  2. Identifying the original coordinates of point B: B(4, -2).
  3. Applying the translation: Subtract 1 from the x-coordinate (for the leftward shift) and add 3 to the y-coordinate (for the upward shift).
  4. Calculating the new coordinates of B': (4 - 1, -2 + 3) = (3, 1).
  5. Comparing the result with the answer choices and selecting the correct answer: C. (3, 1).

This problem highlights the importance of understanding coordinate geometry and geometric transformations. It also demonstrates how to apply these concepts to solve practical problems. The ability to visualize translations and their effects on coordinates is a valuable skill in mathematics and related fields. By mastering these concepts, you can tackle a wide range of geometric problems and gain a deeper appreciation for the connection between algebra and geometry. This problem serves as a good example of how mathematical concepts can be applied to solve real-world problems. Whether you're working on geometric designs, computer graphics, or engineering projects, understanding translations and other transformations is essential. Remember to always break down problems into smaller, manageable steps, and carefully apply the relevant concepts and formulas. With practice and a solid understanding of the fundamentals, you can confidently solve a variety of mathematical challenges. This concludes our exploration of the rectangle ABCD translation problem. We hope this step-by-step solution has been helpful and informative.