Finding Coordinates After Translation A Detailed Explanation

Introduction

In coordinate geometry, transformations play a crucial role in understanding how geometric figures can be manipulated in a plane. Among these transformations, translation is a fundamental concept that involves shifting a figure from one location to another without changing its size or orientation. This article delves into the specifics of translation, particularly focusing on how to determine the coordinates of a point after a translation has been applied. We will explore the mathematical principles behind translations and apply these principles to solve a specific problem: If a translation of $(x, y) ightarrow (x+6, y-10)$ is applied to figure $ABCD$, what are the coordinates of $D'$ if $D$ is $(-5, -2)$? This question serves as an excellent example to illustrate the practical application of translation rules in coordinate geometry.

Understanding Translation in Coordinate Geometry

In coordinate geometry, a translation is a transformation that moves every point of a figure or a space by the same distance in a given direction. A translation can be visualized as sliding a figure without rotating or reflecting it. Mathematically, a translation is defined by a translation vector, which specifies the amount and direction of the shift. This concept is crucial for understanding various geometric transformations and their effects on figures in a coordinate plane. The translation vector essentially dictates how much each point in the figure will move horizontally and vertically. For instance, a translation vector of (a, b) indicates that each point will be moved 'a' units horizontally and 'b' units vertically. If 'a' is positive, the movement is to the right; if negative, it's to the left. Similarly, if 'b' is positive, the movement is upwards; if negative, it's downwards. Understanding these conventions is vital for accurately predicting the new coordinates of a point after a translation.

The Translation Rule

The translation rule is a mathematical expression that describes how the coordinates of a point change under a translation. In general, if a point $(x, y)$ is translated by a vector $(a, b)$, the image of the point, denoted as $(x', y')$, is given by the rule:

$$(x', y') = (x + a, y + b)$$

This rule signifies that the x-coordinate of the image is obtained by adding 'a' to the original x-coordinate, and the y-coordinate of the image is obtained by adding 'b' to the original y-coordinate. This simple yet powerful rule is the cornerstone of performing translations in coordinate geometry. To illustrate, consider a point (2, 3) being translated by the vector (4, -1). Applying the translation rule, the new coordinates would be (2 + 4, 3 - 1), which simplifies to (6, 2). This demonstrates how the translation rule allows us to precisely determine the new location of any point after a translation.

Applying the Translation Rule to Figure ABCD

Given the translation $(x, y) ightarrow (x+6, y-10)$, this rule tells us that every point in the figure $ABCD$ will be moved 6 units to the right (since +6 is added to the x-coordinate) and 10 units downwards (since -10 is added to the y-coordinate). This consistent shift across all points of the figure ensures that the shape and size of $ABCD$ remain unchanged, only its position in the coordinate plane is altered. Understanding this, we can now focus on the specific point $D$ and determine its new location after the translation.

Problem Statement: Finding the Coordinates of D'

We are given that the translation rule is $(x, y) ightarrow (x+6, y-10)$ and the coordinates of point $D$ are $(-5, -2)$. Our task is to find the coordinates of $D'$, which is the image of $D$ after the translation. This involves applying the translation rule to the coordinates of $D$. The problem is a classic example of how translation rules are used in coordinate geometry to determine the new positions of points after a transformation. By correctly applying the rule, we can accurately find the coordinates of $D'$ and understand how the point has moved in the coordinate plane.

Step-by-Step Solution

To find the coordinates of $D'$, we apply the translation rule $(x, y) ightarrow (x+6, y-10)$ to the coordinates of $D$, which are $(-5, -2)$.

1. Apply the Translation to the x-coordinate: The x-coordinate of $D$ is -5. According to the translation rule, we add 6 to this value:

   $$x' = x + 6 = -5 + 6 = 1$$

   This calculation shows that the new x-coordinate of $D'$ will be 1.
2. Apply the Translation to the y-coordinate: The y-coordinate of $D$ is -2. According to the translation rule, we subtract 10 from this value:

   $$y' = y - 10 = -2 - 10 = -12$$

   This calculation shows that the new y-coordinate of $D'$ will be -12.

Therefore, the coordinates of $D'$ are $(1, -12)$. This step-by-step approach ensures that we accurately apply the translation rule to both the x and y coordinates, leading to the correct image of the point after the transformation.

Visualizing the Translation

To further understand this translation, visualize point $D$ at $(-5, -2)$ on the coordinate plane. The translation $(x, y) ightarrow (x+6, y-10)$ moves $D$ six units to the right and ten units down. This movement results in $D'$ being located at $(1, -12)$. Visualizing the translation in this way helps to reinforce the understanding of how the translation rule affects the position of points in the coordinate plane. It also provides a geometric intuition for the algebraic calculations, making the concept more accessible and easier to remember.

Conclusion

In conclusion, by applying the translation rule $(x, y) ightarrow (x+6, y-10)$ to point $D(-5, -2)$, we found that the coordinates of $D'$ are $(1, -12)$. This exercise demonstrates the practical application of translation rules in coordinate geometry. Understanding these transformations is fundamental for more advanced geometric concepts and problem-solving. The ability to accurately apply translation rules is a key skill in mathematics, particularly in areas such as geometry, computer graphics, and spatial reasoning. This problem not only reinforces the understanding of translation but also highlights the importance of precise application of mathematical rules to achieve accurate results.

Final Answer

The final answer is A. (1, -12).