To find out which transformation was applied, let's examine the original square and its coordinates, as well as the transformed square and its coordinates:
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Original square vertices:
- A(3, 3)
- B(6, 3)
- C(6, 6)
- D(3, 6)
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Transformed square vertices:
- A'(-3, -3)
- B'(-6, -3)
- C'(-6, -6)
- D'(-3, -6)
Now let's analyze the transformations:
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Translation:
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Option A: Translation 6 units left and 6 units down would change each point (x, y) to (x - 6, y - 6).
- For point A(3, 3): (3 - 6, 3 - 6) → (-3, -3), which matches A'(-3, -3).
- Checking B, C, and D would yield:
- B(6, 3) → (0, -3) (not matching)
- C(6, 6) → (0, 0) (not matching)
- D(3, 6) → (-3, 0) (not matching)
- So, this option does not work.
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Option B: Translation 6 units right and 6 units up would change each point (x, y) to (x + 6, y + 6).
- This would not result in the transformed points, so it does not match.
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Rotation:
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Option C: Rotation 180 degrees about the origin would result in (x, y) becoming (-x, -y).
- For point A(3, 3): (-3, -3) matches A'(-3, -3).
- Checking other points:
- B(6, 3) → (-6, -3) (matches B'(-6, -3))
- C(6, 6) → (-6, -6) (matches C'(-6, -6))
- D(3, 6) → (-3, -6) (matches D'(-3, -6))
- All points match, so this option works.
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Option D: Rotation 90 degrees clockwise about the origin would change points according to the rule (x, y) → (y, -x), which wouldn’t yield the transformed coordinates.
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Given this analysis, the correct transformation applied is:
C. rotation 180 degrees about the origin.