Given that one vertex of a polygon is located at $(3,-2)$ and after a rotation it is located at $(2,3)$, what is the angle and direction of the rotation about the origin?

• A. A rotation of $180°$
• B. A rotation of $90°$ clockwise
• C. A rotation of $270°$ counterclockwise
• D. A rotation of $90°$ counterclockwise

Answer: (D)

Verified step by step guidance

1. Identify the original point and its image after rotation. The original point is \( (3, -2) \) and the rotated point is \( (2, 3) \).
2. Recall the standard rotation formulas about the origin: - For a rotation of \( \theta \) degrees counterclockwise, the point \( (x, y) \) maps to \( (x', y') = (x \cos{\theta} - y \sin{\theta}, x \sin{\theta} + y \cos{\theta}) \).
3. Test common rotation angles (90°, 180°, 270°) by substituting \( \theta \) into the formulas and checking which angle transforms \( (3, -2) \) into \( (2, 3) \).
4. Alternatively, use the known rotation transformations: - 90° counterclockwise: \( (x, y) \to (-y, x) \) - 180°: \( (x, y) \to (-x, -y) \) - 270° counterclockwise (or 90° clockwise): \( (x, y) \to (y, -x) \)
5. Compare the rotated coordinates with these transformations to determine the angle and direction of rotation that maps \( (3, -2) \) to \( (2, 3) \).