https://file.aiquickdraw.com/mathbot/file/1729458579935y9tgvsec.jpg

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?

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**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.

Find the coordinates of the original points P, Q, R, S before a 180° rotation, given their rotated coordinates on a coordinate plane.

Learn how to find the original coordinates of points after a 180° rotation on a coordinate plane. This problem involves the geometric transformation of a quadrilateral with points P, Q, R, and S. Explore the rotation formula and step-by-step solution.

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