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!