Give me some problems with rotating shapes with ansewrs 

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.

Give me some problems with rotating shapes with answers

Explore problems involving the rotation of various shapes, including triangles, rectangles, circles, and polygons. This page provides solutions and explanations for each rotation scenario, focusing on geometry concepts and the use of rotation matrices. Suitable for Grades 7-10.

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