To determine the image of the point $(-7, 5)$ after a $180^\circ$ counterclockwise rotation about the origin, we use the following rule for a $180^\circ$ rotation:

$(x, y) \to (-x, -y)$

Applying the rule:

Given the point $(-7, 5)$:

• The $x$-coordinate becomes $-(-7) = 7$,
• The $y$-coordinate becomes $-(5) = -5$.

Result:

The image of the point after the rotation is:

$(7, -5)$

Would you like additional details or clarification?

Here are 5 related questions to expand your understanding:

1. How do other rotation angles, such as $90^\circ$ or $270^\circ$, transform points?
2. How would this change if the rotation occurred about a different point (not the origin)?
3. What are the properties of transformations like rotations (e.g., do they preserve distance)?
4. How would the transformation affect a geometric shape rather than just a point?
5. What is the relationship between clockwise and counterclockwise rotations?

Tip: Always sketch a coordinate grid for visualization—it makes rotations much easier to comprehend!