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The problem you uploaded involves a 270° counterclockwise rotation of a triangle \( \triangle DEF \) around the origin, resulting in \( \triangle D'E'F' \).

Let's break this down:

### (a) Mapping the coordinates after rotation

To determine the new coordinates after a 270° counterclockwise rotation about the origin, we can use the following rule:
- **270° counterclockwise rotation**: \((x, y) \rightarrow (y, -x)\)

Using this rule, we apply it to each of the points:

1. **Point D**: 
   - Original coordinates: \( D(-8, 1) \)
   - After applying the rule: \( D' = (1, 8) \)

2. **Point E**: 
   - Original coordinates: \( E(-6, 7) \)
   - After applying the rule: \( E' = (7, 6) \)

3. **Point F**: 
   - Original coordinates: \( F(-2, -3) \)
   - After applying the rule: \( F' = (-3, 2) \)

### (b) General rule for the rotation

As stated earlier, the general rule for a 270° counterclockwise rotation is:
\[
(x, y) \rightarrow (y, -x)
\]

So the correct option for the general rule is:
- \( (x, y) \rightarrow (y, -x) \)

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Would you like any further clarification on how to apply rotation rules? Here are some related questions to expand on this concept:

1. What would the coordinates be after a 90° counterclockwise rotation of \( \triangle DEF \)?
2. What is the formula for rotating a point 180° around the origin?
3. Can you explain how to visualize a 270° rotation on the coordinate plane?
4. What is the difference between rotating clockwise and counterclockwise?
5. How would the coordinates change if rotated 270° clockwise?

**Tip:** Always remember, the rule for 270° counterclockwise rotation is the same as 90° clockwise rotation!

A triangle DEF is rotated 270° counterclockwise about the origin. The result is triangle D'E'F'. The user is asked to fill in the final coordinates of points D', E', and F' after the rotation and choose the correct rule for the rotation.

This problem involves rotating triangle DEF 270° counterclockwise around the origin and determining the coordinates of D', E', and F'. It uses the rule (x, y) → (y, -x) and covers key concepts in geometric transformations. Suitable for students in Grades 8-10.

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