Possible Vertical Translations

Oct 13, 2025 by windailys.com 139 views

Introduction

Translation is a fundamental concept in geometry that involves moving a figure from one location to another without changing its size or shape. In the coordinate plane, translation can be represented by adding or subtracting values to the x and y coordinates of the original figure. In this article, we will explore how to translate a square on the coordinate plane and determine the possible ordered pairs that represent another vertex of the image.

Translation of a Square

A square with side lengths of 4 units is translated on the coordinate plane. One vertex of the image is at ${$ (12, ?)$}$. To find the possible ordered pairs that represent another vertex of the image, we need to understand the concept of translation.

Translation Rules

When a figure is translated on the coordinate plane, the following rules apply:

The x-coordinate of the translated figure is the sum of the x-coordinate of the original figure and the horizontal translation value.
The y-coordinate of the translated figure is the sum of the y-coordinate of the original figure and the vertical translation value.

Finding the Ordered Pair

Since one vertex of the image is at ${$ (12, ?)$}$, we can assume that the original square was translated horizontally by 12 units. To find the possible ordered pairs that represent another vertex of the image, we need to consider the possible vertical translations.

Possible Vertical Translations

Since the side length of the square is 4 units, the possible vertical translations are:

Up by 4 units: ${$ (12, 4)$}$
Down by 4 units: ${$ (12, -4)$}$
Up by 8 units: ${$ (12, 8)$}$
Down by 8 units: ${$ (12, -8)$}$

Analyzing the Options

Now, let's analyze the options provided:

A. ${$ (0, -1)$}$: This option does not match any of the possible vertical translations.
B. ${$ (2, 6)$}$: This option does not match any of the possible vertical translations.
C. ${$ (12, 4)$}$: This option matches the possible vertical translation of up by 4 units.

Conclusion

Based on the analysis, the possible ordered pair that represents another vertex of the image is ${$ (12, 4)$}$. This option matches the possible vertical translation of up by 4 units, which is consistent with the translation rules.

Understanding Translation on the Coordinate Plane: Key Takeaways

Translation is a fundamental concept in geometry that involves moving a figure from one location to another without changing its size or shape.
The x-coordinate of the translated figure is the sum of the x-coordinate of the original figure and the horizontal translation value.
The y-coordinate of the translated figure is the sum of the y-coordinate of the original figure and the vertical translation value.
The possible vertical translations are up by 4 units, down by 4 units, up by 8 units, and down by 8 units.

Practice Problems

A square with side lengths of 5 units is translated on the coordinate plane. One vertex of the image is at ${$ (8, 3)$}$. Find the possible ordered pairs that represent another vertex of the image.
A rectangle with side lengths of 6 units and 8 units is translated on the coordinate plane. One vertex of the image is at ${$ (4, 2)$}$. Find the possible ordered pairs that represent another vertex of the image.

Solutions

The possible vertical translations are up by 5 units, down by 5 units, up by 10 units, and down by 10 units. The possible ordered pairs that represent another vertex of the image are ${$ (8, 8)$}$, ${$ (8, -2)$}$, ${$ (8, 13)$}$, and ${$ (8, -7)$}$.
The possible vertical translations are up by 6 units, down by 6 units, up by 12 units, and down by 12 units. The possible ordered pairs that represent another vertex of the image are ${$ (4, 8)$}$, ${$ (4, -4)$}$, ${$ (4, 14)$}$, and ${$ (4, -10)$}$.

Conclusion

Q: What is the definition of translation in geometry?

A: Translation is a fundamental concept in geometry that involves moving a figure from one location to another without changing its size or shape.

Q: How do you represent translation on the coordinate plane?

A: On the coordinate plane, translation can be represented by adding or subtracting values to the x and y coordinates of the original figure.

Q: What are the translation rules for the x and y coordinates?

A: The x-coordinate of the translated figure is the sum of the x-coordinate of the original figure and the horizontal translation value. The y-coordinate of the translated figure is the sum of the y-coordinate of the original figure and the vertical translation value.

Q: How do you find the possible ordered pairs that represent another vertex of the image?

A: To find the possible ordered pairs that represent another vertex of the image, you need to consider the possible vertical translations. Since the side length of the square is 4 units, the possible vertical translations are up by 4 units, down by 4 units, up by 8 units, and down by 8 units.

Q: What is the possible ordered pair that represents another vertex of the image?

A: Based on the analysis, the possible ordered pair that represents another vertex of the image is ${$ (12, 4)$}$. This option matches the possible vertical translation of up by 4 units, which is consistent with the translation rules.

Q: What are some practice problems to help understand translation on the coordinate plane?

A: Here are some practice problems to help you understand translation on the coordinate plane:

A square with side lengths of 5 units is translated on the coordinate plane. One vertex of the image is at ${$ (8, 3)$}$. Find the possible ordered pairs that represent another vertex of the image.
A rectangle with side lengths of 6 units and 8 units is translated on the coordinate plane. One vertex of the image is at ${$ (4, 2)$}$. Find the possible ordered pairs that represent another vertex of the image.

Q: What are the solutions to the practice problems?

A: Here are the solutions to the practice problems:

The possible vertical translations are up by 5 units, down by 5 units, up by 10 units, and down by 10 units. The possible ordered pairs that represent another vertex of the image are ${$ (8, 8)$}$, ${$ (8, -2)$}$, ${$ (8, 13)$}$, and ${$ (8, -7)$}$.
The possible vertical translations are up by 6 units, down by 6 units, up by 12 units, and down by 12 units. The possible ordered pairs that represent another vertex of the image are ${$ (4, 8)$}$, ${$ (4, -4)$}$, ${$ (4, 14)$}$, and ${$ (4, -10)$}$.

Q: What are some key takeaways from this article?

A: Here are some key takeaways from this article:

Translation is a fundamental concept in geometry that involves moving a figure from one location to another without changing its size or shape.
The x-coordinate of the translated figure is the sum of the x-coordinate of the original figure and the horizontal translation value.
The y-coordinate of the translated figure is the sum of the y-coordinate of the original figure and the vertical translation value.
The possible vertical translations are up by 4 units, down by 4 units, up by 8 units, and down by 8 units.

Q: What are some common mistakes to avoid when working with translation on the coordinate plane?

A: Here are some common mistakes to avoid when working with translation on the coordinate plane:

Failing to consider the possible vertical translations.
Not understanding the translation rules for the x and y coordinates.
Not analyzing the options carefully before selecting the correct answer.

Q: How can you apply the concepts of translation on the coordinate plane to real-world problems?

A: The concepts of translation on the coordinate plane can be applied to real-world problems in various fields, such as:

Architecture: Translation can be used to design and build structures that are symmetrical and aesthetically pleasing.
Engineering: Translation can be used to design and build machines and mechanisms that are efficient and effective.
Computer Science: Translation can be used to develop algorithms and programs that are efficient and effective.

Conclusion

In conclusion, translation is a fundamental concept in geometry that involves moving a figure from one location to another without changing its size or shape. By understanding the translation rules and analyzing the possible vertical translations, we can determine the possible ordered pairs that represent another vertex of the image.