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 \[$ P' \$\]?A.

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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) β†’ (x-2, y-16). We will focus on finding the y-value of P', the translated vertex of P.

Understanding Translation

Translation is a transformation that moves a point or a shape from one location to another. It is a rigid motion, meaning that the size and orientation of the shape remain unchanged. The translation rule (x, y) β†’ (x-2, y-16) indicates that each point (x, y) is moved 2 units to the left and 16 units down.

Applying the Translation Rule

To find the translated vertex P', we need to apply the translation rule to the coordinates of P. The coordinates of P are (-2, 6). We will substitute these values into the translation rule:

(x, y) β†’ (x-2, y-16)

(-2, 6) β†’ (-2-2, 6-16)

(-2, 6) β†’ (-4, -10)

Therefore, the translated vertex P' has coordinates (-4, -10).

Finding the y-Value of P'

The question asks for the y-value of P'. From the previous step, we found that the coordinates of P' are (-4, -10). The y-value of P' is the second coordinate, which is -10.

Conclusion

In this 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) β†’ (x-2, y-16). We applied the translation rule to find the coordinates of the translated vertex P', and then extracted the y-value of P'. The y-value of P' is -10.

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) β†’ (x-2, y-16) indicates that each point (x, y) is moved 2 units to the left and 16 units down.
  • To find the translated vertex P', we need to apply the translation rule to the coordinates of P.
  • The y-value of P' is the second coordinate of the translated vertex P'.

Example Problems

  1. Find the translated vertex Q' of Q(-8,4) according to the rule (x, y) β†’ (x-2, y-16).
  2. Find the translated vertex R' of R(1,-2) according to the rule (x, y) β†’ (x-2, y-16).

Solutions

  1. Q' = (-8-2, 4-16) = (-10, -12)
  2. R' = (1-2, -2-16) = (-1, -18)

Practice Problems

  1. Find the translated vertex P' of P(-2,6) according to the rule (x, y) β†’ (x+2, y+16).
  2. Find the translated vertex Q' of Q(-8,4) according to the rule (x, y) β†’ (x+2, y+16).

Solutions

  1. P' = (-2+2, 6+16) = (0, 22)
  2. Q' = (-8+2, 4+16) = (-6, 20)

Conclusion

Q: What is the definition 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.

Q: What is the translation rule (x, y) β†’ (x-2, y-16) used for?

A: The translation rule (x, y) β†’ (x-2, y-16) is used to move each point (x, y) 2 units to the left and 16 units down.

Q: How do you find the translated vertex P' of P(-2,6) according to the rule (x, y) β†’ (x-2, y-16)?

A: To find the translated vertex P', we need to apply the translation rule to the coordinates of P. We substitute the values of P into the translation rule:

(x, y) β†’ (x-2, y-16)

(-2, 6) β†’ (-2-2, 6-16)

(-2, 6) β†’ (-4, -10)

Therefore, the translated vertex P' has coordinates (-4, -10).

Q: What is the y-value of P'?

A: The y-value of P' is the second coordinate of the translated vertex P', which is -10.

Q: How do you find the translated vertex Q' of Q(-8,4) according to the rule (x, y) β†’ (x-2, y-16)?

A: To find the translated vertex Q', we need to apply the translation rule to the coordinates of Q. We substitute the values of Q into the translation rule:

(x, y) β†’ (x-2, y-16)

(-8, 4) β†’ (-8-2, 4-16)

(-8, 4) β†’ (-10, -12)

Therefore, the translated vertex Q' has coordinates (-10, -12).

Q: What is the y-value of Q'?

A: The y-value of Q' is the second coordinate of the translated vertex Q', which is -12.

Q: How do you find the translated vertex R' of R(1,-2) according to the rule (x, y) β†’ (x-2, y-16)?

A: To find the translated vertex R', we need to apply the translation rule to the coordinates of R. We substitute the values of R into the translation rule:

(x, y) β†’ (x-2, y-16)

(1, -2) β†’ (1-2, -2-16)

(1, -2) β†’ (-1, -18)

Therefore, the translated vertex R' has coordinates (-1, -18).

Q: What is the y-value of R'?

A: The y-value of R' is the second coordinate of the translated vertex R', which is -18.

Q: What are some common mistakes to avoid when applying the translation rule?

A: Some common mistakes to avoid when applying the translation rule include:

  • Not substituting the correct values into the translation rule
  • Not following the correct order of operations (e.g., subtracting 2 from x before subtracting 16 from y)
  • Not checking the units of the translation (e.g., making sure to move the point 2 units to the left and 16 units down)

Q: How can you practice applying the translation rule?

A: You can practice applying the translation rule by working through example problems, such as finding the translated vertices of points with different coordinates. You can also try creating your own example problems and solving them to test your understanding of the translation rule.

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

A: The translation rule has many real-world applications, including:

  • Computer graphics: The translation rule is used to move objects in computer graphics, such as characters or objects in a video game.
  • Architecture: The translation rule is used to move buildings or other structures in architectural designs.
  • Engineering: The translation rule is used to move machines or other mechanical systems in engineering designs.

Conclusion

In this 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) β†’ (x-2, y-16). We applied the translation rule to find the coordinates of the translated vertices P', Q', and R', and then extracted the y-values of P', Q', and R'. We also answered some common questions about the translation rule and provided some tips for practicing and applying the rule.