https://file.aiquickdraw.com/mathbot/file/1727975495198v0xg0up2.jpg

Let's break down and analyze the problem from the worksheet.

### Problem Breakdown:
- **Task:** Rectangle \( ABCD \) is drawn on the coordinate plane. Ravi uses a transformation to form a new rectangle \( A'B'C'D' \).
- **Part a:** You need to identify two different transformations Ravi could have used to map \( ABCD \) onto \( A'B'C'D' \).
- **Part b:** Ravi claims there's a third transformation that could map \( ABCD \) onto \( A'B'C'D' \). We need to analyze whether Ravi's statement is correct.

### Observations:
- The original rectangle \( ABCD \) and the transformed rectangle \( A'B'C'D' \) appear in two graphs, showing different transformations that resulted in the second rectangle.
  
### Solution to Part a:
1. **Transformation 1: Translation**
   - Looking at the first graph on the left:
     - Rectangle \( ABCD \) is translated (shifted) vertically down by 4 units. This could be a **vertical translation**.
     - If the coordinates of \( A \) were \( (1, 5) \), after the translation, \( A' \) would be at \( (1, 1) \).
   
2. **Transformation 2: Reflection**
   - In the second graph (on the right), it appears that the rectangle \( ABCD \) was reflected across the x-axis. 
   - For example, point \( A(1,5) \) reflects to \( A'(1, -5) \), and similarly, the other points of the rectangle follow this pattern.

### Solution to Part b:
- **Ravi's Statement:**
  - Ravi says a third transformation can map \( ABCD \) to \( A'B'C'D' \). This could be true if Ravi is referring to a **rotation**.
  - If Ravi rotates the original rectangle 180 degrees around the origin, it would map to the new position.
    - For example, a point \( A(1, 5) \) rotated by 180 degrees would map to \( A'(-1, -5) \), following the same for other points.
  
### Conclusion:
Yes, Ravi is correct. A 180-degree **rotation** around the origin would indeed be another transformation that maps \( ABCD \) onto \( A'B'C'D' \).

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Do you want more details or have any further questions? Here are some related questions:

1. How can we describe a translation mathematically?
2. What is the formula for reflecting a point across the x-axis?
3. How can you determine the new coordinates of a point after a 180-degree rotation?
4. How would reflecting across the y-axis affect the coordinates of \( ABCD \)?
5. Can you describe a situation where multiple transformations result in the same image?

**Tip:** When reflecting across the x-axis, remember that only the y-coordinates change signs while the x-coordinates remain the same.

Ravi draws rectangle ABCD in the coordinate plane. He uses a single transformation to form the image rectangle A'B'C'D'. a. What transformation could Ravi have used? Label the vertices of the image to show two different ways Ravi could map rectangle ABCD onto A'B'C'D' and identify each transformation. b. Ravi says there is a third single transformation that maps rectangle ABCD onto rectangle A'B'C'D'. Is Ravi correct? Explain.

Learn how Ravi used transformations like translations, reflections, and rotations to map rectangle ABCD onto A'B'C'D'. Discover two transformations, plus whether a third is possible, in this problem suitable for Grades 8-10.

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