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 $A'B'C'D'$.

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.