To address the given problem of rotating the triangle $\triangle DEF$ 90° counterclockwise about the origin, let’s solve it step by step:

Part (a): Determine the coordinates after rotation

The rule for a 90° counterclockwise rotation about the origin is: $(x, y) \to (-y, x)$

Using this rule:

1. Point $D(1, -5)$:

   • New coordinates: $(-(-5), 1) = (5, 1)$
   • $D'(5, 1)$

2. Point $E(8, 2)$:

   • New coordinates: $(-2, 8)$
   • $E'(-2, 8)$

3. Point $F(6, -4)$:

   • New coordinates: $(4, 6)$
   • $F'(4, 6)$

Part (b): General rule for the rotation

The general rule for the 90° counterclockwise rotation is: $(x, y) \to (-y, x)$

Final Answer:

1. Coordinates after rotation (Part a):

   • $D(1, -5) \to D'(5, 1)$
   • $E(8, 2) \to E'(-2, 8)$
   • $F(6, -4) \to F'(4, 6)$

2. General Rule (Part b):

   • Correct answer: $(x, y) \to (-y, x)$

Let me know if you'd like further clarifications or explanations!

Related Questions to Explore:

1. What is the effect of a 180° rotation about the origin on a point's coordinates?
2. How does a 90° clockwise rotation differ from a 90° counterclockwise rotation in terms of transformations?
3. What is the matrix representation of a 90° counterclockwise rotation?
4. Can you generalize the transformation for a rotation by $\theta$ degrees about the origin?
5. How would the triangle change if reflected instead of rotated?

Tip: Visualizing transformations on graph paper or using software can make these rules intuitive and easier to remember!