Rectangle EFGH Is Translated According To The Rule $T_{-5,9}(x, Y$\]. If The Coordinates Of The Pre-image Of Point H Are $(-2, -3$\], What Are The Coordinates Of $H'$?A. $(7, -8$\] B. $(-7, 6$\] C. $(3,

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 rectangle. We will use the translation rule $T_{-5,9}(x, y)$ to find the coordinates of point H' after the translation of rectangle EFGH.

Understanding Translation

Translation is a transformation that moves a figure from one location to another without changing its size or shape. It is a type of rigid motion that preserves the distance and angle between points. In the context of coordinate geometry, translation can be represented by a vector, which is a mathematical object that has both magnitude and direction.

The Translation Rule

The translation rule $T_{-5,9}(x, y)$ indicates that the translation vector is $(-5, 9)$. This means that for every point $(x, y)$ in the original figure, the corresponding point in the translated figure is $(x - 5, y + 9)$.

Applying the Translation Rule

To find the coordinates of point H', we need to apply the translation rule to the coordinates of point H. The coordinates of point H are given as $(-2, -3)$. Using the translation rule, we can find the coordinates of H' as follows:

$$H' = T_{-5,9}(-2, -3) = (-2 - 5, -3 + 9) = (-7, 6)$$

Conclusion

In this article, we applied the translation rule $T_{-5,9}(x, y)$ to find the coordinates of point H' after the translation of rectangle EFGH. We used the coordinates of point H as the pre-image and found the corresponding coordinates of H' using the translation rule. The result is $H' = (-7, 6)$.

The Translation Rule: A Closer Look

The translation rule $T_{-5,9}(x, y)$ is a specific example of a translation vector. In general, a translation vector can be represented as $(h, k)$, where $h$ and $k$ are the horizontal and vertical components of the vector, respectively. The translation rule can be applied to any point $(x, y)$ in the original figure to find the corresponding point in the translated figure.

Applying the Translation Rule: A Step-by-Step Guide

To apply the translation rule, follow these steps:

1. Identify the translation vector $(h, k)$.
2. Identify the coordinates of the pre-image point $(x, y)$.
3. Apply the translation rule by adding $h$ to the x-coordinate and $k$ to the y-coordinate: $(x + h, y + k)$.

Example: Applying the Translation Rule

Suppose we want to apply the translation rule $T_{-3,4}(x, y)$ to the point $(2, 1)$. Following the steps above, we get:

1. Identify the translation vector: $(-3, 4)$.
2. Identify the coordinates of the pre-image point: $(2, 1)$.
3. Apply the translation rule: $(2 + (-3), 1 + 4) = (-1, 5)$.

Conclusion

In this article, we explored the concept of translation and applied it to a specific problem involving a rectangle. We used the translation rule $T_{-5,9}(x, y)$ to find the coordinates of point H' after the translation of rectangle EFGH. We also provided a step-by-step guide on how to apply the translation rule and an example to illustrate the concept.

The Importance of Translation in Geometry

Translation is a fundamental concept in geometry that has numerous applications in various fields, including art, architecture, and engineering. It is used to create symmetries, reflect images, and transform shapes. In this article, we have seen how translation can be used to find the coordinates of a point after a translation.

Real-World Applications of Translation

Translation has numerous real-world applications, including:

• Art and Design: Translation is used to create symmetries and reflect images in art and design.
• Architecture: Translation is used to design buildings and structures that are symmetrical and aesthetically pleasing.
• Engineering: Translation is used to design and build machines and mechanisms that require precise movements and transformations.

Conclusion

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Q: What is translation in geometry?

A: Translation is a transformation that moves a figure from one location to another without changing its size or shape. It is a type of rigid motion that preserves the distance and angle between points.

Q: How is translation represented in coordinate geometry?

A: In coordinate geometry, translation is represented by a vector, which is a mathematical object that has both magnitude and direction. The translation rule $T_{h,k}(x, y)$ indicates that the translation vector is $(h, k)$.

Q: What is the translation rule?

A: The translation rule $T_{h,k}(x, y)$ is a specific example of a translation vector. It is used to find the corresponding point in the translated figure by adding $h$ to the x-coordinate and $k$ to the y-coordinate: $(x + h, y + k)$.

Q: How do I apply the translation rule?

A: To apply the translation rule, follow these steps:

1. Identify the translation vector $(h, k)$.
2. Identify the coordinates of the pre-image point $(x, y)$.
3. Apply the translation rule by adding $h$ to the x-coordinate and $k$ to the y-coordinate: $(x + h, y + k)$.

Q: What is the difference between translation and rotation?

A: Translation and rotation are both types of transformations, but they differ in the way they move a figure. Translation moves a figure from one location to another without changing its size or shape, while rotation turns a figure around a fixed point without changing its size or shape.

Q: Can you provide an example of applying the translation rule?

A: Suppose we want to apply the translation rule $T_{-3,4}(x, y)$ to the point $(2, 1)$. Following the steps above, we get:

1. Identify the translation vector: $(-3, 4)$.
2. Identify the coordinates of the pre-image point: $(2, 1)$.
3. Apply the translation rule: $(2 + (-3), 1 + 4) = (-1, 5)$.

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

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

• Art and Design: Translation is used to create symmetries and reflect images in art and design.
• Architecture: Translation is used to design buildings and structures that are symmetrical and aesthetically pleasing.
• Engineering: Translation is used to design and build machines and mechanisms that require precise movements and transformations.

Q: Can you explain the importance of translation in geometry?

A: Translation is a fundamental concept in geometry that has numerous applications in various fields, including art, architecture, and engineering. It is used to create symmetries, reflect images, and transform shapes.

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

A: To find the coordinates of a point after a translation, you can use the translation rule $T_{h,k}(x, y)$, which adds $h$ to the x-coordinate and $k$ to the y-coordinate: $(x + h, y + k)$.

Q: What is the relationship between translation and the coordinate plane?

A: Translation is a fundamental concept in the coordinate plane, as it allows us to move points and figures from one location to another without changing their size or shape. The translation rule $T_{h,k}(x, y)$ is used to find the corresponding point in the translated figure by adding $h$ to the x-coordinate and $k$ to the y-coordinate: $(x + h, y + k)$.