The image you've uploaded shows triangle $\Delta DEF$ with coordinates for points D, E, and F, along with instructions to graph the triangle and its image after a 180° rotation about the origin.

To perform a 180° rotation about the origin, the coordinates of any point $(x, y)$ transform to $(-x, -y)$.

Let's go through this step-by-step:

• Point D: From the image, the coordinates for $D$ are $(4, -2)$.

  • After a 180° rotation, the new coordinates for D will be $(-4, 2)$.

• Point E: The coordinates for $E$ are $(-1, 4)$.

  • After the rotation, the new coordinates for E will be $(1, -4)$.

• Point F: The coordinates for $F$ are $(2, 1)$.

  • After the rotation, the new coordinates for F will be $(-2, -1)$.

Now, you can graph the triangle with these new points, forming the image after rotation.

Would you like help with further details on how to plot or additional questions on this topic?

Questions to explore further:

1. How can you check if the rotation is correct using geometric properties?
2. How do you apply a 180° rotation to a point that lies on a different quadrant?
3. What happens to the orientation of the triangle after the rotation?
4. How would the image change if a different angle of rotation, like 90° or 270°, was applied?
5. Can this method be used to rotate other polygons? If so, how?

Tip: To ensure accuracy when rotating by 180°, always remember that the points' coordinates simply change sign (i.e., $(x, y)$ becomes $(-x, -y)$).