https://file.aiquickdraw.com/mathbot/file/1730827516214nu8w02s4.jpg

To solve this problem, we need to determine the image of point \( A \) located at \( (-2, 1) \) after a \( 270^\circ \) clockwise rotation about the point \( (3, -3) \).

Here’s the step-by-step approach:

1. **Translate the Points**: 
   Shift the entire setup so that the center of rotation, \( (3, -3) \), moves to the origin. This is done by translating both points by \((-3, +3)\):
   - New coordinates for \( A \) after translation:
     \[
     (-2 - 3, 1 + 3) = (-5, 4)
     \]

2. **Rotate \( 270^\circ \) Clockwise**:
   A \( 270^\circ \) clockwise rotation around the origin is equivalent to a \( 90^\circ \) counterclockwise rotation. The transformation rule for a \( 90^\circ \) counterclockwise rotation is:
   \[
   (x, y) \rightarrow (-y, x)
   \]
   Applying this to \( (-5, 4) \):
   \[
   (-5, 4) \rightarrow (-4, -5)
   \]

3. **Translate Back**:
   Now, move the rotated point back to the original position by translating it by \((+3, -3)\):
   \[
   (-4 + 3, -5 - 3) = (-1, -8)
   \]

Thus, the image of point \( A \) after a \( 270^\circ \) clockwise rotation about \( (3, -3) \) is \( (-1, -8) \).

Would you like more details on any step, or do you have questions?

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Here are 5 related questions you might explore:

1. How would the result change with a \( 90^\circ \) clockwise rotation instead?
2. What is the effect of rotating a point \( 180^\circ \) about a given point?
3. How can rotation transformations be applied to a line segment or shape instead of a single point?
4. How does translating affect the coordinates before and after rotation?
5. Can rotation transformations be applied to three-dimensional coordinates?

**Tip:** When working with rotation, translation can simplify the process by making the rotation center the origin. This makes applying rotation rules straightforward.

Point A is graphed at (-2, 1). Using the pencil, plot the image of point A after a 270° clockwise rotation about (3, -3).

Discover how to find the image of point A at (-2, 1) after a 270° clockwise rotation around the point (3, -3). This problem explores rotation transformations in the coordinate plane, using translation and rotation formulas. Suitable for students in Grades 9-10.

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