Rectangle ABCD Has Vertices \[$A (-6,-2), B (-3,-2), C(-3,-6),\$\] And \[$D(-6,-6)\$\]. The Rectangle Is Translated So That The Coordinates Of The Image Are \[$A'(-10,1), B'(-7,1), C'(-7,-3),\$\] And

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Introduction

In mathematics, translation is a fundamental concept in geometry that involves moving a shape from one position to another without changing its size or orientation. In this article, we will explore the concept of rectangle translation using a specific example. We will analyze the vertices of a given rectangle, translate it to a new position, and determine the coordinates of the image.

Given Rectangle ABCD

The given rectangle ABCD has vertices A(-6,-2), B(-3,-2), C(-3,-6), and D(-6,-6). To understand the translation process, let's visualize the rectangle in a coordinate plane.

A (-6,-2)
B (-3,-2)
C (-3,-6)
D (-6,-6)

Translation of Rectangle ABCD

The rectangle is translated so that the coordinates of the image are A'(-10,1), B'(-7,1), C'(-7,-3), and D'(-10,-3). To find the translation vector, we need to determine the difference between the original and image coordinates.

A' (-10,1)
B' (-7,1)
C' (-7,-3)
D' (-10,-3)

Translation Vector

The translation vector can be found by subtracting the original coordinates from the image coordinates. Let's calculate the translation vector for each vertex.

Δx = x' - x
Δy = y' - y

Δx_A = -10 - (-6) = -4
Δy_A = 1 - (-2) = 3

Δx_B = -7 - (-3) = -4
Δy_B = 1 - (-2) = 3

Δx_C = -7 - (-3) = -4
Δy_C = -3 - (-6) = 3

Δx_D = -10 - (-6) = -4
Δy_D = -3 - (-6) = 3

As we can see, the translation vector is (-4, 3) for all vertices. This means that the rectangle has been translated 4 units to the left and 3 units upwards.

Translation Matrix

The translation matrix can be represented as:

| 1  0  Δx |
| 0  1  Δy |
| 0  0  1  |

In this case, the translation matrix is:

| 1  0  -4 |
| 0  1  3  |
| 0  0  1  |

Conclusion

In this article, we have explored the concept of rectangle translation using a specific example. We have analyzed the vertices of a given rectangle, translated it to a new position, and determined the coordinates of the image. We have also calculated the translation vector and translation matrix. This example demonstrates the importance of understanding translation in mathematics, particularly in geometry.

Applications of Rectangle Translation

Rectangle translation has numerous applications in various fields, including:

  • Computer-Aided Design (CAD): Translation is used to move objects in a design to a new position without changing their size or orientation.
  • Computer Graphics: Translation is used to move objects in a 2D or 3D space to create animations or simulations.
  • Architecture: Translation is used to move buildings or structures to a new location without changing their size or orientation.
  • Engineering: Translation is used to move machines or mechanisms to a new position without changing their size or orientation.

Real-World Examples

Rectangle translation has numerous real-world examples, including:

  • Moving a house: When moving a house to a new location, the translation vector and translation matrix are used to determine the new position of the house.
  • Designing a building: When designing a building, translation is used to move the building to a new location without changing its size or orientation.
  • Creating animations: When creating animations, translation is used to move objects in a 2D or 3D space to create realistic movements.

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions about rectangle translation.

Q: What is rectangle translation?

A: Rectangle translation is a fundamental concept in mathematics that involves moving a rectangle from one position to another without changing its size or orientation.

Q: How do I find the translation vector?

A: To find the translation vector, you need to subtract the original coordinates from the image coordinates. The translation vector is the difference between the original and image coordinates.

Q: What is the translation matrix?

A: The translation matrix is a 3x3 matrix that represents the translation operation. It is used to move a rectangle from one position to another without changing its size or orientation.

Q: How do I use the translation matrix?

A: To use the translation matrix, you need to multiply the original coordinates by the translation matrix. The resulting coordinates will be the image coordinates.

Q: What are the applications of rectangle translation?

A: Rectangle translation has numerous applications in various fields, including:

  • Computer-Aided Design (CAD): Translation is used to move objects in a design to a new position without changing their size or orientation.
  • Computer Graphics: Translation is used to move objects in a 2D or 3D space to create animations or simulations.
  • Architecture: Translation is used to move buildings or structures to a new location without changing their size or orientation.
  • Engineering: Translation is used to move machines or mechanisms to a new position without changing their size or orientation.

Q: What are some real-world examples of rectangle translation?

A: Some real-world examples of rectangle translation include:

  • Moving a house: When moving a house to a new location, the translation vector and translation matrix are used to determine the new position of the house.
  • Designing a building: When designing a building, translation is used to move the building to a new location without changing its size or orientation.
  • Creating animations: When creating animations, translation is used to move objects in a 2D or 3D space to create realistic movements.

Q: How do I calculate the translation vector and translation matrix?

A: To calculate the translation vector and translation matrix, you need to follow these steps:

  1. Determine the original coordinates of the rectangle.
  2. Determine the image coordinates of the rectangle.
  3. Subtract the original coordinates from the image coordinates to find the translation vector.
  4. Create a 3x3 translation matrix using the translation vector.
  5. Multiply the original coordinates by the translation matrix to find the image coordinates.

Q: What are some common mistakes to avoid when using rectangle translation?

A: Some common mistakes to avoid when using rectangle translation include:

  • Not calculating the translation vector correctly: Make sure to subtract the original coordinates from the image coordinates to find the translation vector.
  • Not creating the translation matrix correctly: Make sure to create a 3x3 translation matrix using the translation vector.
  • Not multiplying the original coordinates by the translation matrix correctly: Make sure to multiply the original coordinates by the translation matrix to find the image coordinates.

Conclusion

In conclusion, rectangle translation is a fundamental concept in mathematics that has numerous applications in various fields. By understanding rectangle translation, you can create realistic designs, animations, and simulations. By following the steps outlined in this article, you can calculate the translation vector and translation matrix and avoid common mistakes.