The problem asks for the coordinates of the vertices after a 180° counterclockwise rotation around the origin. Here's how to solve it:

Step-by-step explanation:

1. Rule for 180° Counterclockwise Rotation: When you rotate a point $(x, y)$ by 180° counterclockwise around the origin, the new coordinates will be $(-x, -y)$. This is because a 180° rotation flips both the x- and y-coordinates.

2. Identify the coordinates of the original vertices (from the graph):

   • $R = (9, 8)$
   • $S = (12, 8)$
   • $T = (12, 4)$
   • $Q = (9, 4)$

3. Apply the rotation rule:

   • For $R(9, 8)$, the new coordinates will be $(-9, -8)$.
   • For $S(12, 8)$, the new coordinates will be $(-12, -8)$.
   • For $T(12, 4)$, the new coordinates will be $(-12, -4)$.
   • For $Q(9, 4)$, the new coordinates will be $(-9, -4)$.

Final coordinates after rotation:

• $R' = (-9, -8)$
• $S' = (-12, -8)$
• $T' = (-12, -4)$
• $Q' = (-9, -4)$

Let me know if you'd like further explanation or have any questions!

Here are 5 related questions to explore:

1. What would the coordinates be after a 90° counterclockwise rotation?
2. How would the graph look after a 270° clockwise rotation?
3. What is the general rule for rotating points by 180° clockwise?
4. How does the distance between points change after a 180° rotation?
5. What are the transformations involved in a reflection over the origin?

Tip: When performing a rotation, always pay attention to the center of rotation (origin or any other point) and the angle (90°, 180°, etc.) to apply the correct transformation.