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:

• $(x, y) \rightarrow (y, -x)$

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!