Translation of a Square RSTU

Oct 13, 2025 by recoveryshake.com 116 views

Introduction

In coordinate geometry, translation is a fundamental concept that involves shifting a figure from one position to another without changing its size or orientation. In this article, we will explore the translation of a square in the coordinate plane. We will examine the translation of a square RSTU to form R'S'T'U', and analyze the coordinates of the vertices of the translated square.

Translation of a Square RSTU

The square RSTU is translated to form R'S'T'U', which has vertices ${$ R'(-8, 1) $}$, ${$ S'(-4, 1) $}$, ${$ T'(-4, -3) $}$, and ${$ U'(-8, -3) $}$. To understand the translation, let's first analyze the coordinates of the vertices of the original square RSTU.

Coordinates of the Original Square RSTU

The coordinates of the vertices of the original square RSTU are:

R: ${$ (0, 0) $}$
S: ${$ (6, 0) $}$
T: ${$ (6, -6) $}$
U: ${$ (0, -6) $}$

Translation of the Square RSTU

The square RSTU is translated to form R'S'T'U', which has vertices ${$ R'(-8, 1) $}$, ${$ S'(-4, 1) $}$, ${$ T'(-4, -3) $}$, and ${$ U'(-8, -3) $}$. To find the translation vector, we need to find the difference between the coordinates of the corresponding vertices of the original and translated squares.

Translation Vector

The translation vector can be found by subtracting the coordinates of the corresponding vertices of the original and translated squares. Let's find the translation vector for each vertex:

R'(-8, 1) - R(0, 0) = ${$ (-8, 1) $}$
S'(-4, 1) - S(6, 0) = ${$ (-10, 1) $}$
T'(-4, -3) - T(6, -6) = ${$ (-10, -9) $}$
U'(-8, -3) - U(0, -6) = ${$ (-8, -9) $}$

Analysis of the Translation Vector

The translation vector is the same for all vertices, which means that the square RSTU is translated by the same amount in the x and y directions. The translation vector is ${$ (-10, -9) $}$, which means that the square RSTU is translated 10 units to the left and 9 units downwards.

Point S

Point S has coordinates of ${$ (3, -5) $}$. To find the coordinates of point S' after translation, we need to add the translation vector to the coordinates of point S.

Coordinates of Point S'

The coordinates of point S' can be found by adding the translation vector to the coordinates of point S:

S' = S + translation vector
S' = (3, -5) + (-10, -9)
S' = (-7, -14)

Conclusion

In this article, we analyzed the translation of a square RSTU to form R'S'T'U'. We found the coordinates of the vertices of the translated square and analyzed the translation vector. We also found the coordinates of point S' after translation. The translation vector is ${$ (-10, -9) $}$, which means that the square RSTU is translated 10 units to the left and 9 units downwards.

Translation of a Square in Coordinate Geometry: Key Takeaways

The translation of a square involves shifting the figure from one position to another without changing its size or orientation.
The translation vector is the same for all vertices of the square.
The translation vector can be found by subtracting the coordinates of the corresponding vertices of the original and translated squares.
The coordinates of the vertices of the translated square can be found by adding the translation vector to the coordinates of the vertices of the original square.

References

[1] Coordinate Geometry by Michael Artin
[2] Geometry: A Comprehensive Guide by David A. Brannan
[3] Coordinate Geometry: A Detailed Explanation by Khan Academy
Translation of a Square in Coordinate Geometry: Q&A =====================================================

Introduction

In our previous article, we explored the translation of a square in the coordinate plane. We analyzed the translation of a square RSTU to form R'S'T'U', and examined the coordinates of the vertices of the translated square. In this article, we will answer some frequently asked questions about the translation of a square in coordinate geometry.

Q&A

Q: What is the translation of a square in coordinate geometry?

A: The translation of a square in coordinate geometry involves shifting the figure from one position to another without changing its size or orientation.

Q: How do I find the translation vector of a square?

A: To find the translation vector of a square, you need to subtract the coordinates of the corresponding vertices of the original and translated squares.

Q: What is the significance of the translation vector in coordinate geometry?

A: The translation vector is the same for all vertices of the square, and it represents the amount by which the square is translated in the x and y directions.

Q: How do I find the coordinates of the vertices of the translated square?

A: To find the coordinates of the vertices of the translated square, you need to add the translation vector to the coordinates of the vertices of the original square.

Q: What is the relationship between the translation vector and the coordinates of the vertices of the translated square?

A: The coordinates of the vertices of the translated square can be found by adding the translation vector to the coordinates of the vertices of the original square.

Q: Can a square be translated in any direction?

A: No, a square can only be translated in the x and y directions. It cannot be translated in the z direction.

Q: What is the effect of translation on the size and orientation of a square?

A: Translation does not change the size or orientation of a square. It only shifts the figure from one position to another.

Q: Can a square be translated to any position in the coordinate plane?

A: Yes, a square can be translated to any position in the coordinate plane.

Q: How do I determine the translation vector of a square if I know the coordinates of the vertices of the original and translated squares?

A: To determine the translation vector of a square, you need to subtract the coordinates of the corresponding vertices of the original and translated squares.

Q: What is the significance of the translation vector in real-world applications?

A: The translation vector is used in various real-world applications, such as computer graphics, game development, and engineering.

Conclusion

In this article, we answered some frequently asked questions about the translation of a square in coordinate geometry. We hope that this article has provided you with a better understanding of the translation of a square and its significance in coordinate geometry.

References

[1] Coordinate Geometry by Michael Artin
[2] Geometry: A Comprehensive Guide by David A. Brannan
[3] Coordinate Geometry: A Detailed Explanation by Khan Academy