https://file.aiquickdraw.com/mathbot/file/17272941130384jmzrke7.jpg

To solve this problem, we need to find the new position of the point \( P(-7, 2) \) after a counterclockwise rotation of \( 270^\circ \) about the origin.

### Step-by-Step Process:

1. **Understanding Rotation**:
   A rotation of \( 270^\circ \) counterclockwise is equivalent to a \( 90^\circ \) clockwise rotation. We can use this concept to help visualize the rotation.

2. **Rotation Rule**:
   When a point \( (x, y) \) is rotated by \( 270^\circ \) counterclockwise, the new coordinates \( (x', y') \) are given by:
   \[
   (x', y') = (y, -x)
   \]
   This is the general transformation rule for a \( 270^\circ \) counterclockwise rotation.

3. **Apply the Rotation**:
   - The original point is \( (-7, 2) \).
   - After applying the rule, the new coordinates are:
     \[
     P' = (2, 7)
     \]

4. **Determine the Quadrant**:
   - The point \( (2, 7) \) has a positive x-coordinate and a positive y-coordinate.
   - This means that \( P' \) is located in **Quadrant I**.

Thus, the correct answer is **Option \#1: Quadrant I**.

Would you like any more details on this? Here are some related questions to explore:

1. How do you determine the new coordinates of a point after a \( 180^\circ \) rotation?
2. What are the effects of a \( 90^\circ \) clockwise rotation on a point?
3. Can you generalize the transformation rules for any angle of rotation?
4. What quadrant would a point lie in after a \( 360^\circ \) rotation?
5. What happens to the distance of a point from the origin after a rotation?

**Tip**: Rotations about the origin always preserve the distance between a point and the origin.

P(-7, 2) is rotated 270° counterclockwise about the origin. Name the quadrant P' is located in. Enter the option number that corresponds to your answer.

Learn how to determine the quadrant where the point P(-7, 2) is located after a 270° counterclockwise rotation about the origin. This problem involves applying the rotation transformation rule for 270°, and understanding the quadrant system in coordinate geometry. Suitable for students in Grades 9-10.

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