To analyze the transformations and the incorrect transformation rule mentioned, let's break this down into the required steps: translating the points by \( (2,1) \) and then rotating them \( 90 \) degrees clockwise.
Given Points
- A (1, 1)
- B (2, 4)
- C (4, 2)
- D (3, 1)
- E (2, 2)
Step 1: Translate the Points by (2, 1)
To translate a point \( (x, y) \) by \( (2, 1) \), we add \( 2 \) to \( x \) and \( 1 \) to \( y \).
- A: \( (1+2, 1+1) = (3, 2) \)
- B: \( (2+2, 4+1) = (4, 5) \)
- C: \( (4+2, 2+1) = (6, 3) \)
- D: \( (3+2, 1+1) = (5, 2) \)
- E: \( (2+2, 2+1) = (4, 3) \)
Translated points:
- A (3, 2)
- B (4, 5)
- C (6, 3)
- D (5, 2)
- E (4, 3)
Step 2: Rotate 90 Degrees Clockwise
To rotate a point \( (x, y) \) \( 90 \) degrees clockwise, we use the transformation: \[ (x, y) \rightarrow (y, -x) \]
Applying this to our translated points:
- A (3, 2): \( (2, -3) \)
- B (4, 5): \( (5, -4) \)
- C (6, 3): \( (3, -6) \)
- D (5, 2): \( (2, -5) \)
- E (4, 3): \( (3, -4) \)
Rotated points:
- A (2, -3)
- B (5, -4)
- C (3, -6)
- D (2, -5)
- E (3, -4)
Comparing with Given After Points
The given "After" points were:
- A' (-2, 3)
- B' (-5, 4)
- C' (-3, 6)
- D' (-2, 5)
- E' (-3, 4)
Incorrect Transformation Rule Analysis
The incorrect transformation rule suggested was to rotate \( 180 \) degrees counterclockwise instead of performing the correct steps.
180 degrees counterclockwise rotation: The transformation for a point \( (x, y) \) rotating \( 180 \) degrees is given by: \[ (x, y) \rightarrow (-x, -y) \]
Let's apply this to the original points before translation:
- A (1, 1): \( (-1, -1) \)
- B (2, 4): \( (-2, -4) \)
- C (4, 2): \( (-4, -2) \)
- D (3, 1): \( (-3, -1) \)
- E (2, 2): \( (-2, -2) \)
This incorrect transformation does not yield the points after transformation specified in the problem.
Summary
The correct transformation process yields points different from what was provided as "After" points. The incorrect transformation mentioned (rotating \( 180 \) degrees counterclockwise) shows how vital it is to apply transformations in the sequence and the type dictated in the problem statement. The final points obtained through the correct method do not match the provided "After" points, indicating a potential error in the expected results or required transformations beyond simple translation and rotation.