The transformation rule $(x, y) \to (-x, -y)$ means that each point of the figure is transformed into its opposite in both the $x$- and $y$-directions. This corresponds to rotating the figure $180^\circ$ about the origin.

Explanation:

• A $180^\circ$ rotation about the origin flips each point directly across the origin. For instance:
  • The point $(a, b)$ becomes $(-a, -b)$.
  • The point $(-a, -b)$ becomes $(a, b)$.

None of the other options are correct:

• A $90^\circ$ rotation clockwise: This would map $(x, y)$ to $(y, -x)$, which is different.
• Reflection across the $x$-axis: This maps $(x, y)$ to $(x, -y)$, which does not match the rule.
• Reflection across the $y$-axis: This maps $(x, y)$ to $(-x, y)$, which also does not match the rule.

Correct Answer:

B. A $180^\circ$ clockwise rotation about the origin.

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Do you want further clarification or a diagram for this transformation?

Related Questions:

1. What is the difference between rotation and reflection in geometry?
2. How can we use transformation rules to identify reflections?
3. What happens when you apply the rule $(x, y) \to (-x, -y)$ twice to a point?
4. How can coordinate rules help in predicting compositions of transformations?
5. Can you describe transformations that result from other angles of rotation ($90^\circ, 270^\circ$)?

Tip:

When identifying transformations, always test the rule on specific points to understand its effect on the figure.