If A is rotated 180° about the origin, what are the coordinates Of A' show work

To determine the coordinates of \( A' \), the image of point \( A \) after a 180° rotation about the origin, we follow this process:

### Formula for 180° Rotation
When a point \( (x, y) \) is rotated 180° about the origin, its new coordinates become:
\[
(x', y') = (-x, -y)
\]

### Example Calculation
Let \( A(x, y) \) represent the original coordinates of the point. Applying the formula:
- Original coordinates of \( A \): \( (x, y) \)
- After 180° rotation about the origin:
\[
A'(x', y') = (-x, -y)
\]

### Step-by-Step Work:
1. Take the original coordinates \( (x, y) \).
2. Negate the x-coordinate to get \( -x \).
3. Negate the y-coordinate to get \( -y \).

Thus, \( A'(-x, -y) \) is the rotated point.

### Example:
If \( A(3, 4) \):
\[
A' = (-3, -4)
\]

If \( A(-2, 5) \):
\[
A' = (2, -5)
\]

### General Conclusion:
The transformation flips both the x and y coordinates to their negatives.

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Do you want to calculate for specific coordinates? Let me know! 

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### Related Questions:
1. What happens if we rotate a point 90° instead of 180°?
2. Can we find the coordinates of a point after 360° rotation?
3. How do 180° rotations about a point other than the origin work?
4. How can we visualize a 180° rotation on a graph?
5. What is the relationship between 180° rotation and reflection through the origin?

### Tip:
When rotating a point about the origin, always check the signs of both coordinates to ensure accuracy.

If A is rotated 180° about the origin, what are the coordinates of A'? Show work.

Discover how to calculate the coordinates of a point after a 180° rotation about the origin. This detailed guide explains the transformation formula, includes step-by-step examples, and covers key concepts in geometry and coordinate transformations. Ideal for Grades 8-10.

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