Hexagon DEFGHI Is Translated 8 Units Down And 3 Units To The Right. If The Coordinates Of The Pre-image Of Point F Are \[$(-9,2)\$\], What Are The Coordinates Of \[$F^*\$\]?A. \[$(-12,-6)\$\] B. \[$(-6,-6)\$\] C.

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Introduction

In geometry, coordinate translations are a fundamental concept used to describe the movement of points and shapes in a two-dimensional plane. A translation is a transformation that moves a point or shape from one location to another without changing its size or orientation. In this article, we will explore how to apply coordinate translations to find the new coordinates of a point after a given translation.

What is a Coordinate Translation?

A coordinate translation is a transformation that moves a point or shape from one location to another by adding or subtracting a fixed value to its x-coordinate and y-coordinate. This can be represented as:

(x, y) → (x + h, y + k)

where (x, y) is the original point, (x + h, y + k) is the new point, and (h, k) is the translation vector.

Applying Coordinate Translations

To apply a coordinate translation, we need to add the translation vector to the original coordinates of the point. Let's consider the given problem:

Hexagon DEFGHI is translated 8 units down and 3 units to the right.

This means that the translation vector is (3, -8).

Finding the Coordinates of F*

The coordinates of the pre-image of point F are (-9, 2). To find the coordinates of F*, we need to add the translation vector to the original coordinates of F.

F* = (-9 + 3, 2 - 8) F* = (-6, -6)

Therefore, the coordinates of F* are (-6, -6).

Conclusion

In this article, we have explored how to apply coordinate translations to find the new coordinates of a point after a given translation. We have used the concept of translation vectors to move a point from one location to another. By following the steps outlined in this article, you can apply coordinate translations to solve a wide range of geometry problems.

Example Problems

  1. A point is translated 4 units up and 2 units to the left. If the original coordinates of the point are (3, -2), what are the new coordinates of the point?
  2. A hexagon is translated 6 units down and 1 unit to the right. If the coordinates of the pre-image of point F are (-5, 3), what are the coordinates of F*?
  3. A triangle is translated 3 units up and 2 units to the left. If the original coordinates of the triangle are (2, -1), what are the new coordinates of the triangle?

Solutions

  1. The new coordinates of the point are (1, -2 + 4) = (1, 2).
  2. The coordinates of F* are (-5 + 1, 3 - 6) = (-4, -3).
  3. The new coordinates of the triangle are (2 - 2, -1 + 3) = (0, 2).

Tips and Tricks

  • When applying a coordinate translation, make sure to add the translation vector to the original coordinates of the point.
  • Use the concept of translation vectors to move a point from one location to another.
  • Practice applying coordinate translations to solve a wide range of geometry problems.

Glossary

  • Coordinate translation: A transformation that moves a point or shape from one location to another without changing its size or orientation.
  • Translation vector: A vector that represents the movement of a point or shape in a two-dimensional plane.
  • Pre-image: The original point or shape before a transformation is applied.
  • Image: The new point or shape after a transformation is applied.
    Coordinate Translations Q&A =============================

Frequently Asked Questions

Q: What is a coordinate translation?

A: A coordinate translation is a transformation that moves a point or shape from one location to another without changing its size or orientation. It is represented by adding or subtracting a fixed value to its x-coordinate and y-coordinate.

Q: How do I apply a coordinate translation?

A: To apply a coordinate translation, you need to add the translation vector to the original coordinates of the point. The translation vector is represented as (h, k), where h is the change in the x-coordinate and k is the change in the y-coordinate.

Q: What is a translation vector?

A: A translation vector is a vector that represents the movement of a point or shape in a two-dimensional plane. It is represented as (h, k), where h is the change in the x-coordinate and k is the change in the y-coordinate.

Q: What is the difference between a pre-image and an image?

A: The pre-image is the original point or shape before a transformation is applied, while the image is the new point or shape after a transformation is applied.

Q: How do I find the coordinates of the image after a coordinate translation?

A: To find the coordinates of the image after a coordinate translation, you need to add the translation vector to the original coordinates of the pre-image.

Q: What are some common types of coordinate translations?

A: Some common types of coordinate translations include:

  • Translation by a fixed distance in the x-direction
  • Translation by a fixed distance in the y-direction
  • Translation by a fixed distance in both the x and y directions

Q: How do I determine the translation vector for a given coordinate translation?

A: To determine the translation vector for a given coordinate translation, you need to identify the change in the x-coordinate and the change in the y-coordinate.

Q: What are some real-world applications of coordinate translations?

A: Some real-world applications of coordinate translations include:

  • Computer graphics and animation
  • Video game development
  • Architecture and engineering
  • Navigation and mapping

Q: How do I practice applying coordinate translations?

A: You can practice applying coordinate translations by working through examples and exercises, such as:

  • Translating points and shapes by a fixed distance in the x-direction
  • Translating points and shapes by a fixed distance in the y-direction
  • Translating points and shapes by a fixed distance in both the x and y directions

Q: What are some common mistakes to avoid when applying coordinate translations?

A: Some common mistakes to avoid when applying coordinate translations include:

  • Failing to add the translation vector to the original coordinates of the pre-image
  • Failing to identify the change in the x-coordinate and the change in the y-coordinate
  • Failing to use the correct translation vector for the given coordinate translation

Q: How do I check my work when applying coordinate translations?

A: You can check your work by:

  • Verifying that the translation vector is correct
  • Verifying that the coordinates of the image are correct
  • Checking that the transformation is correct

Q: What are some resources for learning more about coordinate translations?

A: Some resources for learning more about coordinate translations include:

  • Online tutorials and videos
  • Textbooks and workbooks
  • Online courses and lectures
  • Practice problems and exercises