Which of the following sets of transformations will map rectangle onto itself?
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
0. Review of College Algebra
Transformations
Multiple Choice
Given that one vertex of a polygon is located at and after a rotation it is located at , what is the angle and direction of the rotation about the origin?
A
A rotation of
B
A rotation of clockwise
C
A rotation of counterclockwise
D
A rotation of counterclockwise
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Verified step by step guidance1
Identify the original point and its image after rotation. The original point is \( (3, -2) \) and the rotated point is \( (2, 3) \).
Recall the standard rotation formulas about the origin:
- For a rotation of \( \theta \) degrees counterclockwise, the point \( (x, y) \) maps to \( (x', y') = (x \cos{\theta} - y \sin{\theta}, x \sin{\theta} + y \cos{\theta}) \).
Test common rotation angles (90°, 180°, 270°) by substituting \( \theta \) into the formulas and checking which angle transforms \( (3, -2) \) into \( (2, 3) \).
Alternatively, use the known rotation transformations:
- 90° counterclockwise: \( (x, y) \to (-y, x) \)
- 180°: \( (x, y) \to (-x, -y) \)
- 270° counterclockwise (or 90° clockwise): \( (x, y) \to (y, -x) \)
Compare the rotated coordinates with these transformations to determine the angle and direction of rotation that maps \( (3, -2) \) to \( (2, 3) \).
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