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.