Reflect Quadrilateral PQRS About The Y-axis. Then Translate The Resulting Quadrilateral 2 Units Down. What Are The Coordinates Of The Vertices Of The Final Image? A. P’(12, -6), Q’(8, -7), R’(4, -4), And S‘(7, -1) B. P’(12, 8), Q’(8, 9), R’(4, 6),

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Introduction

In geometry, reflecting and translating shapes are fundamental concepts that help us understand how to manipulate and transform objects in a two-dimensional space. In this article, we will explore the process of reflecting a quadrilateral about the y-axis and then translating the resulting quadrilateral 2 units down. We will also determine the coordinates of the vertices of the final image.

Reflecting a Quadrilateral about the Y-axis

To reflect a quadrilateral about the y-axis, we need to change the sign of the x-coordinate of each vertex while keeping the y-coordinate the same. This is because the y-axis acts as a mirror, and the reflected point will have the same y-coordinate but an opposite x-coordinate.

Let's consider a quadrilateral PQRS with the following coordinates:

  • P(6, 3)
  • Q(8, 2)
  • R(4, 5)
  • S(2, 1)

To reflect this quadrilateral about the y-axis, we will change the sign of the x-coordinate of each vertex:

  • P(6, 3) becomes P'(-6, 3)
  • Q(8, 2) becomes Q'(-8, 2)
  • R(4, 5) becomes R'(-4, 5)
  • S(2, 1) becomes S'(-2, 1)

Translating the Reflected Quadrilateral 2 Units Down

Now that we have reflected the quadrilateral about the y-axis, we need to translate the resulting quadrilateral 2 units down. This means that we will decrease the y-coordinate of each vertex by 2 units.

Using the reflected coordinates from the previous step, we will subtract 2 from the y-coordinate of each vertex:

  • P'(-6, 3) becomes P''(-6, 1)
  • Q'(-8, 2) becomes Q''(-8, 0)
  • R'(-4, 5) becomes R''(-4, 3)
  • S'(-2, 1) becomes S''(-2, -1)

Determining the Coordinates of the Final Image

The final image of the quadrilateral after reflecting it about the y-axis and then translating it 2 units down is the quadrilateral P''Q''R''S''. The coordinates of the vertices of this quadrilateral are:

  • P''(-6, 1)
  • Q''(-8, 0)
  • R''(-4, 3)
  • S''(-2, -1)

Therefore, the correct answer is:

A. P'(12, -6), Q'(8, -7), R'(4, -4), and S‘(7, -1)

This answer is incorrect, as the coordinates of the vertices of the final image are not P'(12, -6), Q'(8, -7), R'(4, -4), and S‘(7, -1). The correct coordinates are P''(-6, 1), Q''(-8, 0), R''(-4, 3), and S''(-2, -1).

Conclusion

Frequently Asked Questions

In the previous article, we explored the process of reflecting a quadrilateral about the y-axis and then translating the resulting quadrilateral 2 units down. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the difference between reflecting and translating a shape?

A: Reflecting a shape involves changing the position of the shape in a way that it appears to be mirrored across a line or axis. Translating a shape, on the other hand, involves moving the shape from one position to another without changing its orientation or size.

Q: How do I reflect a quadrilateral about the x-axis?

A: To reflect a quadrilateral about the x-axis, you need to change the sign of the y-coordinate of each vertex while keeping the x-coordinate the same. This is because the x-axis acts as a mirror, and the reflected point will have the same x-coordinate but an opposite y-coordinate.

Q: What happens if I reflect a quadrilateral about the y-axis and then translate it 3 units up?

A: If you reflect a quadrilateral about the y-axis and then translate it 3 units up, the resulting shape will be the same as if you had translated the original quadrilateral 3 units up. The reflection about the y-axis does not affect the vertical position of the shape.

Q: Can I reflect a quadrilateral about a line that is not an axis?

A: Yes, you can reflect a quadrilateral about a line that is not an axis. However, the process is more complex and involves using the concept of perpendicular bisectors and the midpoint of the line segment connecting the two points.

Q: How do I determine the coordinates of the vertices of the final image after reflecting and translating a quadrilateral?

A: To determine the coordinates of the vertices of the final image, you need to follow the steps of the transformation carefully. First, reflect the quadrilateral about the specified axis or line. Then, translate the resulting shape by the specified amount. Finally, use the new coordinates of the vertices to determine the coordinates of the final image.

Q: What are some real-world applications of reflecting and translating shapes?

A: Reflecting and translating shapes have many real-world applications in fields such as architecture, engineering, and computer graphics. For example, architects use these concepts to design buildings and other structures, while engineers use them to create models of machines and mechanisms. Computer graphics artists use these concepts to create 3D models and animations.

Q: Can I use software or online tools to help me with reflecting and translating shapes?

A: Yes, there are many software programs and online tools available that can help you with reflecting and translating shapes. Some popular options include graphing calculators, geometry software, and online geometry tools.

Conclusion

In this article, we have answered some frequently asked questions related to reflecting and translating a quadrilateral. We hope that this information has been helpful in clarifying some of the concepts and procedures involved in these geometric transformations. If you have any further questions or need additional clarification, please don't hesitate to ask.