An image is drawn from a figure. After a 180 degree(Counterclockwise) rotation about the origin, P(7,-2) is mapped to P' (-7,2). Identify the rule that maps P to P'.

To determine the rule that maps point \( P(7, -2) \) to \( P'(-7, 2) \) after a 180-degree counterclockwise rotation about the origin, we use the general rule for such a rotation.

### General Rule for 180-Degree Counterclockwise Rotation:

When a point \( (x, y) \) is rotated 180 degrees counterclockwise around the origin, the new coordinates \( (x', y') \) are given by:

\[
(x', y') = (-x, -y)
\]

### Applying the Rule:

Given point \( P(7, -2) \):
- After applying the 180-degree rotation rule:
\[
P'(x', y') = (-7, 2)
\]

This matches the provided mapping \( P \to P' \). Thus, the rule that maps \( P \) to \( P' \) under a 180-degree counterclockwise rotation is:

\[
(x, y) \to (-x, -y)
\]

This confirms that each coordinate is negated after the rotation.

Would you like to see more examples of rotations, or do you have other questions?

### Related Questions:

1. What is the rule for a 90-degree counterclockwise rotation around the origin?
2. How does a 270-degree counterclockwise rotation differ from a 90-degree clockwise rotation?
3. How can you determine the coordinates of a point after a reflection over the y-axis?
4. What is the effect of a 360-degree rotation on a point's coordinates?
5. How does rotation affect the distance between two points from the origin?

**Tip:** Remember, rotation transformations preserve distances but change the orientation of the points relative to the center of rotation.

Understand how to map points using a 180-degree counterclockwise rotation around the origin with an example problem. Learn the rule that maps point P(7, -2) to P'(-7, 2) through negating coordinates. Suitable for Grades 9-12.

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