Translation Of Square ABCD Finding The Y-Coordinate Of B'

Jun 25, 2025 by ADMIN 58 views

If a translation \$\$\tau_{-3,-8}(x, y)\$\$ is applied to a square \$\$ABCD\$\$, what is the \$\$y\$\$-coordinate of \$\$B'\$\$ after the transformation?

$If a Translation of \$\$\tau_{-3,-8}(x, y)\$\$ is Applied to Square \$\$ABCD\$\$, What is the \$\$y\$\$-Coordinate of \$\$B'\$\$?$

Understanding Transformations and Coordinate Geometry

In the realm of coordinate geometry, transformations play a pivotal role in manipulating geometric figures within a coordinate plane. These transformations, which include translations, rotations, reflections, and dilations, alter the position or size of a figure while preserving certain geometric properties. Among these transformations, translations hold a special significance due to their ability to shift figures without altering their shape or orientation. This article delves into the intricacies of translations, particularly focusing on how they affect the coordinates of points in a figure. We will explore the concept of translating a square, a fundamental geometric shape, and determine the resulting coordinates of its vertices after the translation. This exploration will provide a comprehensive understanding of how translations work and their impact on geometric figures in the coordinate plane. The given problem presents a scenario where a square, denoted as ABCD, undergoes a translation defined by the vector $\tau_{-3,-8}(x, y)$. This notation signifies a translation that shifts every point in the square 3 units to the left (in the negative x-direction) and 8 units downward (in the negative y-direction). The objective is to determine the y-coordinate of the transformed vertex B', which is the image of vertex B after the translation. To solve this problem, we need to understand the effect of translations on coordinates and apply this understanding to the specific scenario presented. This involves identifying the original coordinates of vertex B, applying the translation vector to these coordinates, and then extracting the y-coordinate of the resulting point, B'. This process will not only provide the answer to the problem but also reinforce the fundamental principles of coordinate geometry and transformations. The question is, if we apply the translation $\tau_{-3,-8}(x, y)$ to the square $ABCD$, what will be the $y$-coordinate of the new point $B'$? We are given the following options:

A. -12 B. -8 C. -6 D. -2

Let's break down the problem and find the correct answer.

Understanding the Translation

The translation $\tau_{-3,-8}(x, y)$ represents a shift in the coordinate plane. The notation $\tau_{a, b}(x, y)$ generally means that every point $(x, y)$ is moved to a new point $(x + a, y + b)$. In our case, $a = -3$ and $b = -8$. This means each point is shifted 3 units to the left (because of the -3) and 8 units down (because of the -8).

Key Concepts of Translations

A translation is a rigid transformation, meaning it preserves the size and shape of the figure being transformed. Only the position changes.
Translations are defined by a translation vector, which specifies the direction and magnitude of the shift.
To apply a translation to a point, simply add the components of the translation vector to the coordinates of the point.

Analyzing the Square ABCD

We don't have the specific coordinates of the square $ABCD$, but we can still solve the problem conceptually. Let's assume that the coordinates of point $B$ are $(x, y)$. After the translation $\tau_{-3,-8}(x, y)$, the new coordinates of point $B'$ will be:

$B' = (x + (-3), y + (-8)) = (x - 3, y - 8)$

Our goal is to find the $y$-coordinate of $B'$, which is $y - 8$. To determine the specific value of $y - 8$, we need additional information about the original coordinates of point $B$. However, the answer choices provided suggest that the solution is independent of the original $x$ and $y$ coordinates of point $B$, but only consider the translation vector $\tau_{-3,-8}(x, y)$. This indicates that the problem is designed to test our understanding of how translations affect coordinates in general.

The Importance of Understanding Coordinate Transformations

Coordinate transformations are a fundamental concept in mathematics and have wide-ranging applications in various fields, including computer graphics, robotics, and physics. Understanding how transformations affect geometric figures is crucial for solving problems in these areas. For instance, in computer graphics, transformations are used to manipulate objects on the screen, such as rotating, scaling, and translating them. In robotics, transformations are used to control the movement of robots and their end-effectors. In physics, transformations are used to describe the motion of objects in space.

Focusing on the Y-Coordinate

The translation affects the $y$-coordinate by subtracting 8. So, if the original $y$-coordinate of $B$ is $y$, the new $y$-coordinate of $B'$ is $y - 8$. The question ultimately asks us for the $y$-coordinate change due to the translation, not the final $y$-coordinate itself.

Visualizing the Translation

To better understand the effect of the translation, imagine a coordinate plane with the square ABCD plotted on it. The translation $\tau_{-3,-8}(x, y)$ can be visualized as sliding the entire square 3 units to the left and 8 units down. This means that each vertex of the square will move in the same way, including vertex B. The y-coordinate of B will decrease by 8 units due to the downward shift.

Analyzing the Answer Choices

Now, let's consider the answer choices:

A. -12 B. -8 C. -6 D. -2

Since we know the $y$-coordinate changes by -8 due to the translation, we are looking for the option that represents this change. The options seem to represent potential final $y$-coordinates, but we need to consider the translation's effect on the $y$-coordinate. If we consider B. -8 as the change in the $y$-coordinate due to the translation, this makes sense based on the translation vector.

The Importance of Careful Interpretation

This problem highlights the importance of carefully interpreting the question and the answer choices. It's easy to get confused and try to calculate the final $y$-coordinate of $B'$ directly, but the question is asking for the effect of the translation on the $y$-coordinate. This distinction is crucial for arriving at the correct answer.

The Solution

The translation $\tau_{-3,-8}(x, y)$ shifts the $y$-coordinate by -8. Therefore, the $y$-coordinate of $B'$ will be 8 units less than the $y$-coordinate of $B$. While we don't know the exact coordinates of $B$, the change in the $y$-coordinate is -8.

Therefore, the correct answer is B. -8.

Conclusion

In conclusion, by understanding the principles of coordinate geometry and translations, we can effectively determine the effect of transformations on geometric figures. In this case, the translation $\tau_{-3,-8}(x, y)$ applied to square $ABCD$ results in a shift of 8 units downward in the $y$-coordinate. Therefore, the $y$-coordinate of $B'$ is 8 units less than the $y$-coordinate of $B$, making option B (-8) the correct answer. This problem serves as a valuable exercise in applying the concepts of transformations and coordinate geometry to solve practical problems.