https://file.aiquickdraw.com/mathbot/file/1726680724509p2nopvei.jpg

Let's analyze the problem in the image.

### Part (a): Coordinate Transformation by Rotation
The triangle \( \triangle DEF \) is rotated 180° counterclockwise about the origin to get the new triangle \( \triangle D'E'F' \).

We are given the original coordinates:
- \( D(-5, 4) \)
- \( E(3, 6) \)
- \( F(4, 2) \)

The task is to find the coordinates after the 180° rotation.

#### 180° Rotation Rule:
For any point \((x, y)\), a 180° counterclockwise rotation around the origin maps the point to \((-x, -y)\).

Using this rule, let's find the new coordinates:
- \( D(-5, 4) \rightarrow D'(5, -4) \)
- \( E(3, 6) \rightarrow E'(-3, -6) \)
- \( F(4, 2) \rightarrow F'(-4, -2) \)

### Part (b): General Rule
The general rule for a 180° rotation around the origin is:
\[
(x, y) \rightarrow (-x, -y)
\]
This rule describes how the coordinates are transformed during the rotation.

### Final Answers:
- **(a)** The final coordinates after the rotation are:
  - \( D' (5, -4) \)
  - \( E' (-3, -6) \)
  - \( F' (-4, -2) \)
  
- **(b)** The correct general rule is: 
\[
(x, y) \rightarrow (-x, -y)
\]

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What is the general formula for a 90° rotation about the origin?
2. How would you rotate a point 270° counterclockwise about the origin?
3. What happens to the distances between points after a 180° rotation?
4. How does a 180° rotation affect the orientation of a shape?
5. Can the general rule \( (x, y) \rightarrow (-x, -y) \) be applied for rotations not centered at the origin?

### Tip:
When rotating figures, always visualize or sketch the rotation to ensure that both direction and magnitude of the transformation are correct.

A triangle DEF is rotated 180° counterclockwise about the origin. Determine the final coordinates of points D, E, and F after the rotation, and select the correct general rule for the transformation.

Learn how to rotate triangle DEF 180 degrees counterclockwise around the origin on the coordinate plane. This problem involves understanding the rotation formula (x, y) → (-x, -y) and applying it to find new coordinates.

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