How would you describe the series of transformations that have occurred to move the rectangle in Quadrant IV to the rectangle in Quadrant II to show that the rectangles are congruent?

Question

1. reflection, then translation
2. translation, then rotation
3. rotation, then reflection
4. rotation, then translation

Answer 1

To determine the series of transformations that can move a rectangle in Quadrant IV to a rectangle in Quadrant II while maintaining congruence (meaning the rectangles remain the same size and shape), we need to understand the positions of the quadrants:

• Quadrant IV is where the x-coordinates are positive and the y-coordinates are negative.
• Quadrant II is where the x-coordinates are negative and the y-coordinates are positive.

Looking at the transformations:

1. Reflection: Reflecting the rectangle over the y-axis would change the rectangle's position from Quadrant IV to Quadrant II. This reflects the x-coordinates, making them negative, while the y-coordinates would remain unchanged.

2. Translation: After reflecting, you may translate (move) the rectangle to its new position if necessary. However, reflecting alone is sufficient to move the rectangle into Quadrant II, where its dimensions and angles remain unchanged, thus maintaining congruence.

Based on this understanding, the correct answer is:

1. reflection, then translation.

This correctly describes the series of transformations needed to demonstrate the congruence of the rectangles. The essential transformation that first positions the rectangle correctly from Quadrant IV to Quadrant II is the reflection.