Triangle ABC Is Translated According To The Rule \[$(x, Y) \rightarrow (x+2, Y-8)\$\]. If The Coordinates Of The Pre-image Of Point B Are \[$(4, -5)\$\], What Are The Coordinates Of \[$B^{\prime}\$\]?A. \[$(2, 3)\$\]B.

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Introduction

In geometry, translation is a fundamental concept that involves moving a figure from one location to another without changing its size or shape. In this article, we will explore the concept of translation and apply it to a specific problem involving a triangle ABC. We will use the given translation rule to find the coordinates of point B' after the translation of point B.

Translation Rule

The translation rule given is {(x, y) \rightarrow (x+2, y-8)$}$. This means that for any point (x, y), its image after translation will be (x+2, y-8).

Pre-Image of Point B

The coordinates of the pre-image of point B are given as {(4, -5)$}$. This means that point B is located at (4, -5) before the translation.

Finding the Coordinates of Point B'

To find the coordinates of point B', we need to apply the translation rule to the coordinates of point B. Using the translation rule, we get:

{(4+2, -5-8) = (6, -13)$}$

Therefore, the coordinates of point B' are (6, -13).

Conclusion

In this article, we applied the concept of translation to find the coordinates of point B' after the translation of point B. We used the given translation rule to move point B from its pre-image location to its new location. The coordinates of point B' are (6, -13).

Example Problem

Suppose we have a triangle ABC with point A located at (2, 3), point B located at (4, -5), and point C located at (6, 1). If the triangle is translated according to the rule {(x, y) \rightarrow (x+2, y-8)$}$, what are the coordinates of point A', point B', and point C'?

To solve this problem, we need to apply the translation rule to each point. For point A, we get:

{(2+2, 3-8) = (4, -5)$}$

For point B, we get:

{(4+2, -5-8) = (6, -13)$}$

For point C, we get:

{(6+2, 1-8) = (8, -7)$}$

Therefore, the coordinates of point A' are (4, -5), the coordinates of point B' are (6, -13), and the coordinates of point C' are (8, -7).

Discussion

Translation is an important concept in geometry that involves moving a figure from one location to another without changing its size or shape. In this article, we applied the concept of translation to find the coordinates of point B' after the translation of point B. We used the given translation rule to move point B from its pre-image location to its new location. The coordinates of point B' are (6, -13).

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 translation rule is given by {(x, y) \rightarrow (x+2, y-8)$}$.
  • To find the coordinates of point B', we need to apply the translation rule to the coordinates of point B.
  • The coordinates of point B' are (6, -13).

Glossary

  • Translation: A transformation that involves moving a figure from one location to another without changing its size or shape.
  • Pre-image: The original location of a point or figure before the translation.
  • Image: The new location of a point or figure after the translation.

References

  • [1] "Geometry: A Comprehensive Introduction". By Dan Pedoe. Published by Dover Publications.
  • [2] "Mathematics for Elementary Teachers". By John F. Harper. Published by McGraw-Hill Education.

About the Author

Introduction

In our previous article, we explored the concept of translation and applied it to a specific problem involving a triangle ABC. We used the given translation rule to find the coordinates of point B' after the translation of point B. In this article, we will answer some frequently asked questions related to the concept of translation and its application to triangle ABC.

Q&A

Q: What is translation in geometry?

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

Q: What is the translation rule given in the problem?

A: The translation rule given is {(x, y) \rightarrow (x+2, y-8)$}$. This means that for any point (x, y), its image after translation will be (x+2, y-8).

Q: How do we find the coordinates of point B' after the translation of point B?

A: To find the coordinates of point B', we need to apply the translation rule to the coordinates of point B. Using the translation rule, we get:

{(4+2, -5-8) = (6, -13)$}$

Therefore, the coordinates of point B' are (6, -13).

Q: What is the pre-image of point B?

A: The pre-image of point B is the original location of point B before the translation. In this case, the pre-image of point B is (4, -5).

Q: What is the image of point B after the translation?

A: The image of point B after the translation is point B', which has coordinates (6, -13).

Q: How do we apply the translation rule to find the coordinates of point A', point B', and point C'?

A: To apply the translation rule, we need to add 2 to the x-coordinate and subtract 8 from the y-coordinate of each point. For point A, we get:

{(2+2, 3-8) = (4, -5)$}$

For point B, we get:

{(4+2, -5-8) = (6, -13)$}$

For point C, we get:

{(6+2, 1-8) = (8, -7)$}$

Therefore, the coordinates of point A' are (4, -5), the coordinates of point B' are (6, -13), and the coordinates of point C' are (8, -7).

Q: What is the significance of translation in geometry?

A: Translation is an important concept in geometry that involves moving a figure from one location to another without changing its size or shape. It is used to solve problems involving the movement of objects in a plane.

Q: What are some real-world applications of translation?

A: Translation has many real-world applications, including:

  • Computer graphics: Translation is used to move objects in a 2D or 3D space.
  • Video games: Translation is used to move characters and objects in a game.
  • Architecture: Translation is used to move buildings and structures in a design.
  • Engineering: Translation is used to move machines and mechanisms in a design.

Conclusion

In this article, we answered some frequently asked questions related to the concept of translation and its application to triangle ABC. We hope that this article has provided a better understanding of the concept of translation and its significance in geometry.

Key Takeaways

  • Translation is a transformation that involves moving a figure from one location to another without changing its size or shape.
  • The translation rule is given by {(x, y) \rightarrow (x+2, y-8)$}$.
  • To find the coordinates of point B', we need to apply the translation rule to the coordinates of point B.
  • The coordinates of point B' are (6, -13).
  • Translation has many real-world applications, including computer graphics, video games, architecture, and engineering.

Glossary

  • Translation: A transformation that involves moving a figure from one location to another without changing its size or shape.
  • Pre-image: The original location of a point or figure before the translation.
  • Image: The new location of a point or figure after the translation.
  • Computer graphics: The use of computers to create and manipulate visual images.
  • Video games: Interactive digital games that can be played on a computer or gaming console.
  • Architecture: The art and science of designing buildings and structures.
  • Engineering: The application of scientific and mathematical principles to design and develop solutions to real-world problems.

References

  • [1] "Geometry: A Comprehensive Introduction". By Dan Pedoe. Published by Dover Publications.
  • [2] "Mathematics for Elementary Teachers". By John F. Harper. Published by McGraw-Hill Education.

About the Author

The author is a mathematics educator with a passion for teaching and learning. They have a strong background in geometry and have taught various mathematics courses to students of all ages.