Triangle PQR Has Vertices { P(-2, 6)$}$, { Q(-8, 4)$}$, And { R(1, -2)$}$. It Is Translated According To The Rule { (x, Y) \rightarrow (x-2, Y-16)$}$.What Is The { Y$}$-value Of

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Introduction

In geometry, translation is a fundamental concept that involves moving a point or a shape from one location to another without changing its size or orientation. In this article, we will explore the translation of a triangle PQR with vertices {P(-2, 6)$}$, {Q(-8, 4)$}$, and {R(1, -2)$}$ according to the rule {(x, y) \rightarrow (x-2, y-16)$}$. We will focus on finding the new y-value of the triangle after the translation.

Understanding the Translation Rule

The translation rule {(x, y) \rightarrow (x-2, y-16)$}$ indicates that each point (x, y) is moved to a new location (x-2, y-16). This means that the x-coordinate is shifted 2 units to the left, and the y-coordinate is shifted 16 units downward.

Applying the Translation Rule to the Triangle

To find the new y-value of the triangle, we need to apply the translation rule to each vertex. Let's start with vertex P(-2, 6).

Vertex P(-2, 6)

Using the translation rule, we can find the new coordinates of vertex P as follows:

  • x-coordinate: x - 2 = -2 - 2 = -4
  • y-coordinate: y - 16 = 6 - 16 = -10

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

Vertex Q(-8, 4)

Using the translation rule, we can find the new coordinates of vertex Q as follows:

  • x-coordinate: x - 2 = -8 - 2 = -10
  • y-coordinate: y - 16 = 4 - 16 = -12

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

Vertex R(1, -2)

Using the translation rule, we can find the new coordinates of vertex R as follows:

  • x-coordinate: x - 2 = 1 - 2 = -1
  • y-coordinate: y - 16 = -2 - 16 = -18

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

Conclusion

In this article, we have explored the translation of a triangle PQR with vertices {P(-2, 6)$}$, {Q(-8, 4)$}$, and {R(1, -2)$}$ according to the rule {(x, y) \rightarrow (x-2, y-16)$}$. We have applied the translation rule to each vertex and found the new y-value of the triangle. The new y-value of the triangle is -18.

Key Takeaways

  • Translation is a fundamental concept in geometry that involves moving a point or a shape from one location to another without changing its size or orientation.
  • The translation rule {(x, y) \rightarrow (x-2, y-16)$}$ indicates that each point (x, y) is moved to a new location (x-2, y-16).
  • To find the new y-value of a triangle after translation, we need to apply the translation rule to each vertex.

Further Reading ----------------If you want to learn more about geometry and translation, here are some recommended resources:

  • Khan Academy: Geometry
  • Math Open Reference: Translation
  • Wolfram MathWorld: Translation

References

  • [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [2] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Introduction

In our previous article, we explored the translation of a triangle PQR with vertices {P(-2, 6)$}$, {Q(-8, 4)$}$, and {R(1, -2)$}$ according to the rule {(x, y) \rightarrow (x-2, y-16)$}$. We found the new y-value of the triangle to be -18. In this article, we will answer some frequently asked questions related to the translation of the triangle.

Q&A

Q: What is the purpose of translation in geometry?

A: Translation is a fundamental concept in geometry that involves moving a point or a shape from one location to another without changing its size or orientation. It is used to describe the movement of objects in a two-dimensional or three-dimensional space.

Q: How do I apply the translation rule to a point or a shape?

A: To apply the translation rule, you need to add the translation vector to each point or vertex of the shape. For example, if the translation rule is {(x, y) \rightarrow (x-2, y-16)$}$, you would add (-2, -16) to each point or vertex.

Q: What is the difference between translation and rotation?

A: Translation involves moving a point or a shape from one location to another without changing its size or orientation. Rotation, on the other hand, involves rotating a point or a shape around a fixed point or axis without changing its size or position.

Q: Can I use the translation rule to translate a shape in a three-dimensional space?

A: Yes, you can use the translation rule to translate a shape in a three-dimensional space. However, you need to consider the x, y, and z coordinates of the shape and apply the translation vector to each coordinate.

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

A: To find the new coordinates of a point after translation, you need to add the translation vector to the original coordinates of the point. For example, if the original coordinates of a point are (x, y) and the translation vector is (a, b), the new coordinates of the point will be (x + a, y + b).

Q: Can I use the translation rule to translate a shape with a variable number of vertices?

A: Yes, you can use the translation rule to translate a shape with a variable number of vertices. However, you need to apply the translation vector to each vertex of the shape.

Conclusion

In this article, we have answered some frequently asked questions related to the translation of a triangle PQR with vertices {P(-2, 6)$}$, {Q(-8, 4)$}$, and {R(1, -2)$}$ according to the rule {(x, y) \rightarrow (x-2, y-16)$}$. We have provided explanations and examples to help you understand the concept of translation and how to apply it to different shapes and scenarios.

Key Takeaways

  • Translation is a fundamental concept in geometry that involves moving a point or a shape from one location to another without changing its size or orientation.
  • The translation rule {(x, y) \ (x-2, y-16)$}$ indicates that each point (x, y) is moved to a new location (x-2, y-16).
  • To find the new coordinates of a point after translation, you need to add the translation vector to the original coordinates of the point.

Further Reading ----------------If you want to learn more about geometry and translation, here are some recommended resources:

  • Khan Academy: Geometry
  • Math Open Reference: Translation
  • Wolfram MathWorld: Translation

References

  • [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [2] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for informational purposes only and are not directly related to the content of this article.