In the provided problem, you are asked to perform a 180° rotation of a line segment about the origin. To solve this, here is the step-by-step process for rotating any point or line segment by 180°:

Steps for a 180° rotation about the origin:

1. Identify the coordinates of the endpoints of the line segment.

   • Let the endpoints of the original line segment be $(x_1, y_1)$ and $(x_2, y_2)$.

2. Apply the 180° rotation rule:

   • For any point $(x, y)$, after a 180° rotation about the origin, the new coordinates will be $(-x, -y)$.

3. Find the new coordinates of the endpoints after rotation:

   • If the original coordinates are $(x_1, y_1)$, after rotation, the new point will be $(-x_1, -y_1)$.
   • Similarly, for $(x_2, y_2)$, the new point will be $(-x_2, -y_2)$.

4. Plot the new points on the graph to complete the rotated line segment.

In your graph:

• From the image, the line segment appears to be between approximately the points $(-3, 2)$ and $(-1, 5)$.

Applying the 180° rotation:

• Point $(-3, 2)$ after a 180° rotation becomes $(3, -2)$.
• Point $(-1, 5)$ after a 180° rotation becomes $(1, -5)$.

Final result:

• You should plot the line segment connecting the points $(3, -2)$ and $(1, -5)$ on the graph.

Would you like a further explanation or have questions about any part of the process?

Here are 5 related questions to deepen understanding:

1. What happens to a point under a 90° rotation about the origin?
2. How would the coordinates change for a 270° rotation about the origin?
3. What is the general rule for rotating a point by any angle $\theta$?
4. How do you rotate figures that are not centered on the origin?
5. Can a 180° rotation be considered a reflection in some cases?

Tip: When performing rotations, always remember to change the signs of both coordinates for a 180° rotation!