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To solve the problem of finding the new coordinates after a 180° counterclockwise rotation around the origin, we need to apply a specific transformation to each vertex of the shape. When a point \((x, y)\) is rotated 180° around the origin, its new coordinates become \((-x, -y)\).

Let's identify the coordinates of the vertices \(B, C, D,\) and \(E\) from the given image and calculate their new positions:

### Original Coordinates:
- **B**: \((-2, 4)\)
- **C**: \((2, 6)\)
- **D**: \((2, 8)\)
- **E**: \((-2, 6)\)

### New Coordinates after 180° Rotation:
Applying the transformation \((x, y) \to (-x, -y)\) to each point:

1. **B**: \((-2, 4) \to (2, -4)\)
2. **C**: \((2, 6) \to (-2, -6)\)
3. **D**: \((2, 8) \to (-2, -8)\)
4. **E**: \((-2, 6) \to (2, -6)\)

### Conclusion:
The coordinates of the vertices after a 180° counterclockwise rotation around the origin are:
- **B**: \((2, -4)\)
- **C**: \((-2, -6)\)
- **D**: \((-2, -8)\)
- **E**: \((2, -6)\)

Would you like further details or have any questions?

### Related Questions:
1. What would the coordinates be after a 90° counterclockwise rotation around the origin?
2. How would you find the coordinates after a 270° counterclockwise rotation around the origin?
3. How does rotating shapes affect their orientation and position on a graph?
4. How can you determine the center of rotation for any rotation in a coordinate plane?
5. What is the general formula for rotating a point around the origin by an arbitrary angle \(\theta\)?

### Tip:
Remember that rotating a point 180° around the origin is the same as reflecting it through the origin.

Write the coordinates of the vertices after a rotation 180° counterclockwise around the origin.

This problem demonstrates how to find the new coordinates of a shape's vertices after a 180° counterclockwise rotation around the origin. Using the transformation formula (-x, -y), the coordinates of points B, C, D, and E are updated accordingly. Suitable for students in Grades 8-10.

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