The image shows a geometric figure plotted on a coordinate plane along with its rotated version after a 180° rotation. The original figure consists of points $P$, $Q$, $R$, and $S$, while the rotated image has points labeled $P'$, $Q'$, $R'$, and $S'$. The task is to find the coordinates of the original points.

A 180° rotation about the origin (which I assume is the center of rotation here) flips the coordinates of a point $(x, y)$ to $(-x, -y)$. Using this information, we can reverse the transformation and find the original coordinates from the rotated ones.

Here are the coordinates of the rotated points:

• $P'$ is at approximately $(-1, 5)$
• $Q'$ is at approximately $(-1, 2)$
• $R'$ is at approximately $(3, 2)$
• $S'$ is at approximately $(3, 5)$

To find the original points before the rotation, we reverse the 180° rotation rule:

1. $P$ (original) is the reflection of $P'$: $P = (1, -5)$

2. $Q$ (original) is the reflection of $Q'$: $Q = (1, -2)$

3. $R$ (original) is the reflection of $R'$: $R = (-3, -2)$

4. $S$ (original) is the reflection of $S'$: $S = (-3, -5)$

Thus, the coordinates of the original points $P, Q, R, S$ are:

• $P(1, -5)$
• $Q(1, -2)$
• $R(-3, -2)$
• $S(-3, -5)$

Would you like more details or have any questions?

Further questions to explore:

1. How does the transformation change if the rotation is 90° instead of 180°?
2. Can you describe the pattern that occurs when reflecting a point through different quadrants?
3. How would the figure look after a 270° rotation about the origin?
4. What are the properties of figures that remain unchanged after a 180° rotation?
5. How do you perform a reflection over the y-axis for these points?

Tip: When performing a 180° rotation, always negate both coordinates of a point to find its new position.