Applying the Translation Rule to Vertex P

Oct 13, 2025 by windailys.com 93 views

In mathematics, the concept of translation is a fundamental aspect of geometry, where a figure is moved from one location to another without any change in its size or shape. In this article, we will explore the translation of triangle PQR, which has vertices at points P(-2,6), Q(-8,4), and R(1,-2). We will apply the translation rule (x, y) → (x-2, y-16) to find the new coordinates of the vertices and determine the y-value of the translated triangle.

Translation Rule

The translation rule (x, y) → (x-2, y-16) indicates that for every point (x, y) in the original triangle, we need to subtract 2 from the x-coordinate and subtract 16 from the y-coordinate to obtain the new coordinates. This rule will be applied to each vertex of the triangle to find the new coordinates.

Applying the Translation Rule to Vertex P

The original coordinates of vertex P are (-2, 6). To apply the translation rule, we subtract 2 from the x-coordinate and subtract 16 from the y-coordinate:

(-2 - 2, 6 - 16) = (-4, -10)

Therefore, the new coordinates of vertex P are (-4, -10).

Applying the Translation Rule to Vertex Q

The original coordinates of vertex Q are (-8, 4). To apply the translation rule, we subtract 2 from the x-coordinate and subtract 16 from the y-coordinate:

(-8 - 2, 4 - 16) = (-10, -12)

Therefore, the new coordinates of vertex Q are (-10, -12).

Applying the Translation Rule to Vertex R

The original coordinates of vertex R are (1, -2). To apply the translation rule, we subtract 2 from the x-coordinate and subtract 16 from the y-coordinate:

(1 - 2, -2 - 16) = (-1, -18)

Therefore, the new coordinates of vertex R are (-1, -18).

New Coordinates of the Translated Triangle

After applying the translation rule to each vertex of the triangle, we obtain the new coordinates:

Vertex P: (-4, -10)
Vertex Q: (-10, -12)
Vertex R: (-1, -18)

Determining the y-Value of the Translated Triangle

To determine the y-value of the translated triangle, we need to find the minimum and maximum y-values among the new coordinates of the vertices. The minimum y-value is -18, and the maximum y-value is -10. Therefore, the y-value of the translated triangle is the range of y-values, which is [-18, -10].

Conclusion

In this article, we applied the translation rule (x, y) → (x-2, y-16) to the vertices of triangle PQR and determined the new coordinates of the vertices. We also found the y-value of the translated triangle, which is the range of y-values among the new coordinates of the vertices. This demonstrates the concept of translation in geometry and how it can be applied to find the new coordinates of a figure after it has been moved from one location to another.

Key Takeaways

The translation rule (x, y) → (x-2, y-16) was applied to the vertices of triangle PQR.
The new coordinates of the vertices were determined after applying the translation rule.
The y-value of the translated triangle was found to be the range of y-values among the new coordinates of the vertices.

Further Exploration

This article has provided a basic understanding of the translation of triangle PQR. However, there are many other aspects of geometry that can be explored further, such as:

Rotations: How do rotations affect the coordinates of a figure?
Reflections: How do reflections affect the coordinates of a figure?
Compositions of Transformations: How do multiple transformations affect the coordinates of a figure?

In the previous article, we explored the translation of triangle PQR, which has vertices at points P(-2,6), Q(-8,4), and R(1,-2). We applied the translation rule (x, y) → (x-2, y-16) to find the new coordinates of the vertices and determined the y-value of the translated triangle. In this article, we will answer some frequently asked questions about the translation of triangle PQR.

Q: What is the purpose of translation in geometry?

A: The purpose of translation in geometry is to move a figure from one location to another without any change in its size or shape. This is useful in various applications, such as mapping, design, and architecture.

Q: How do I apply the translation rule to a figure?

A: To apply the translation rule, you need to subtract the translation value from the x-coordinate and subtract the translation value from the y-coordinate. For example, if the translation rule is (x, y) → (x-2, y-16), you would subtract 2 from the x-coordinate and subtract 16 from the y-coordinate.

Q: What is the difference between translation and rotation?

A: Translation and rotation are two different types of transformations in geometry. Translation involves moving a figure from one location to another without any change in its size or shape, while rotation involves rotating a figure around a fixed point without any change in its size or shape.

Q: Can I apply multiple translations to a figure?

A: Yes, you can apply multiple translations to a figure. Each translation will move the figure to a new location, and the new coordinates of the figure will be determined by the translation rule.

Q: How do I determine the y-value of a translated triangle?

A: To determine the y-value of a translated triangle, you need to find the minimum and maximum y-values among the new coordinates of the vertices. The minimum y-value is the lowest y-value among the vertices, and the maximum y-value is the highest y-value among the vertices.

Q: What is the significance of the y-value of a translated triangle?

A: The y-value of a translated triangle is significant because it determines the vertical position of the triangle. A higher y-value indicates that the triangle is located higher on the coordinate plane, while a lower y-value indicates that the triangle is located lower on the coordinate plane.

Q: Can I apply the translation rule to a 3D figure?

A: Yes, you can apply the translation rule to a 3D figure. However, you need to consider the x, y, and z coordinates of the figure and apply the translation rule to each coordinate separately.

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

A: Some real-world applications of translation in geometry include:

Mapping: Translation is used in mapping to move a figure from one location to another without any change in its size or shape.
Design: Translation is used in design to move a figure from one location to another without any change in its size or shape.
Architecture: Translation is used in architecture to move a figure from one location to another without any change in its size or shape.

Conclusion

In this article, we answered some frequently asked questions about the translation of triangle PQR. We explored the purpose of translation in geometry, how to apply the translation rule, and the significance of the y-value of a translated triangle. We also discussed some real-world applications of translation in geometry. By understanding the concept of translation, you can develop a deeper understanding of geometry and improve your problem-solving skills.

Key Takeaways

Translation involves moving a figure from one location to another without any change in its size or shape.
The translation rule (x, y) → (x-2, y-16) was applied to the vertices of triangle PQR.
The y-value of a translated triangle is determined by finding the minimum and maximum y-values among the new coordinates of the vertices.
Translation has many real-world applications, including mapping, design, and architecture.

Further Exploration

This article has provided a basic understanding of the translation of triangle PQR. However, there are many other aspects of geometry that can be explored further, such as:

Rotations: How do rotations affect the coordinates of a figure?
Reflections: How do reflections affect the coordinates of a figure?
Compositions of Transformations: How do multiple transformations affect the coordinates of a figure?

These are just a few examples of the many topics that can be explored in geometry. By continuing to learn and explore these concepts, you can develop a deeper understanding of the subject and improve your problem-solving skills.