Rotation 180, degrees180 
∘
  counterclockwise around the origin of the point left parenthesis, 7, comma, 1, right parenthesis(7,1)

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 \( (-7, -1) \).

Would you like further details, or do you have any questions?

### 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.

Rotation 180 degrees counterclockwise around the origin of the point (7, 1)

Explore how to rotate the point (7, 1) by 180 degrees counterclockwise around the origin. This solution covers rotation transformations in geometry, specifically the application of the 180-degree rotation formula, and is suitable for students in Grades 8-10.

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