https://file.aiquickdraw.com/mathbot/file/1728014065484iztis8qf.jpg

The problem asks to rotate the given triangle 180 degrees counterclockwise around the origin. Let's break this down:

1. **Rotation by 180 degrees (counterclockwise)**: In a 180-degree rotation about the origin, the coordinates of a point \((x, y)\) are transformed to \((-x, -y)\).

Let's apply this to the vertices \(C\), \(D\), and \(E\) of the triangle in the diagram:

- \(C(x_C, y_C)\)
- \(D(x_D, y_D)\)
- \(E(x_E, y_E)\)

By applying the 180-degree rotation, the new coordinates will be:
- \(C'(-x_C, -y_C)\)
- \(D'(-x_D, -y_D)\)
- \(E'(-x_E, -y_E)\)

### Checking the options:
- Look for the option where each point has been reflected across the origin, i.e., both x and y coordinates are negated. Based on the image, **option A** appears to correctly reflect the triangle after the 180-degree counterclockwise rotation.

Would you like more details about the calculation, or have any specific questions about this transformation?

Here are some related questions for further understanding:
1. How do we perform a 90-degree counterclockwise rotation around the origin?
2. What is the formula for reflecting a point across the y-axis?
3. How do you determine the coordinates of a point after a 270-degree clockwise rotation?
4. What happens when you rotate a point by 360 degrees around the origin?
5. How can you verify the correctness of a geometric transformation?

**Tip**: When performing rotations on a coordinate plane, always remember that a 180-degree rotation changes both the x and y signs!

Rotate the following figure 180 degrees counterclockwise.

Learn how to rotate a triangle 180 degrees counterclockwise around the origin on a coordinate plane. This geometry problem covers concepts of rotations and coordinate transformations, with a step-by-step explanation. Suitable for students in Grades 7-9.

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