To rotate the point $(7, 1)$ by $180^\circ$ counterclockwise around the origin, we use the formula for a $180^\circ$ rotation, which transforms any point $(x, y)$ to $(-x, -y)$.

For the point $(7, 1)$:

1. Apply the transformation: $(x, y) \rightarrow (-x, -y)$
2. Substitute $x = 7$ and $y = 1$: $(7, 1) \rightarrow (-7, -1)$

Result: The rotated point is $(-7, -1)$.

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Related Questions

1. What is the formula for a $90^\circ$ counterclockwise rotation around the origin?
2. How would we rotate the point $(-3, 5)$ by $180^\circ$?
3. What happens to the coordinates of any point after a $360^\circ$ rotation?
4. How would a $180^\circ$ clockwise rotation differ in terms of the point's coordinates?
5. Can you generalize a rule for rotating any point by $270^\circ$ counterclockwise?

Tip: For rotations around the origin, rotating $180^\circ$ always results in the point being mapped to its exact opposite quadrant on the coordinate plane.