Here are some problems involving the rotation of shapes, along with their answers:

1. Problem 1: Rotating a Triangle A triangle with vertices at (1, 1), (4, 1), and (2.5, 3) is rotated 90 degrees counterclockwise around the origin. What are the new coordinates of the vertices?

   Answer: The new vertices are (-1, 1), (-1, 4), (-3, 2.5).

2. Problem 2: Rotating a Rectangle A rectangle with corners at (2, 3), (6, 3), (6, 5), and (2, 5) is rotated 180 degrees around the center point (4, 4). What are the new coordinates?

   Answer: The new corners are (6, 5), (2, 5), (2, 3), and (6, 3).

3. Problem 3: Rotating a Circle A circle centered at (0, 0) with a radius of 3 is rotated 270 degrees around the origin. Does the position of the circle change?

   Answer: No, the position of the circle remains the same; it is still centered at (0, 0) with a radius of 3.

4. Problem 4: Rotating a Polygon A pentagon has vertices at (0, 0), (1, 2), (2, 1), (1, -1), and (-1, -2). If it is rotated 45 degrees around the origin, what are the new coordinates of the vertex (1, 2)?

   Answer: The new coordinates of the vertex (1, 2) are approximately (−0.71, 2.12).

5. Problem 5: Rotating a Point A point located at (3, 4) is rotated 90 degrees clockwise around the origin. What are the new coordinates?

   Answer: The new coordinates are (4, -3).

Would you like to delve into any specific problem for more details, or do you have any questions? Here are some related questions you might consider:

1. How do you derive the rotation matrix for a specific angle?
2. What is the effect of rotating a shape multiple times?
3. How does the center of rotation affect the final position of a shape?
4. Can you apply rotation transformations to 3D shapes?
5. How do you rotate shapes in a coordinate system with different axes?

Tip: Always use rotation matrices to simplify calculations when rotating points or shapes.