Questions: What set of reflections and rotations would carry rectangle ABCD onto itself?

What set of reflections and rotations would carry rectangle ABCD onto itself?
Transcript text: What set of reflections and rotations would carry rectangle $A B C D$ onto itself?
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Solution

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Solution Steps

Step 1: Identify the Transformations

To determine which set of reflections and rotations will carry rectangle ABCD onto itself, we need to analyze the given options and understand how each transformation affects the rectangle.

Step 2: Analyze Each Option

  1. Rotate 180°, reflect over the x-axis, reflect over the line y = x:

    • Rotating 180° will place each vertex of the rectangle in the opposite quadrant.
    • Reflecting over the x-axis will flip the rectangle upside down.
    • Reflecting over the line y = x will swap the x and y coordinates of each vertex.

  2. Reflect over the y-axis, rotate 180°, reflect over the x-axis:

    • Reflecting over the y-axis will flip the rectangle horizontally.
    • Rotating 180° will place each vertex in the opposite quadrant.
    • Reflecting over the x-axis will flip the rectangle upside down.

  3. Rotate 180°, reflect over the y-axis, reflect over the line y = x:

    • Rotating 180° will place each vertex in the opposite quadrant.
    • Reflecting over the y-axis will flip the rectangle horizontally.
    • Reflecting over the line y = x will swap the x and y coordinates of each vertex.

  4. Reflect over the y-axis, reflect over the x-axis, rotate 180°:

    • Reflecting over the y-axis will flip the rectangle horizontally.
    • Reflecting over the x-axis will flip the rectangle upside down.
    • Rotating 180° will place each vertex in the opposite quadrant.
Step 3: Determine the Correct Set of Transformations
  • The correct set of transformations should return the rectangle to its original position.
  • Option 1: Rotate 180°, reflect over the x-axis, reflect over the line y = x.

Final Answer

The set of reflections and rotations that would carry rectangle ABCD onto itself is:

  • Rotate 180°, reflect over the x-axis, reflect over the line y = x.
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