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.