To solve the problem shown in the image, which involves rotating a triangle on the coordinate plane about the origin $(0,0)$, let's break it down step by step.

Given Points

• $A(-2, 2)$
• $B(-3, 1)$
• $C(0, 0)$

Rotation Scenarios

1. 90° Counterclockwise Rotation:

   • The rotation rule for 90° counterclockwise about the origin is $(x, y) \rightarrow (-y, x)$.
   • $A(-2, 2)$ becomes $A'(-2, -2)$.
   • $B(-3, 1)$ becomes $B'(-1, -3)$.
   • $C(0, 0)$ remains $C'(0, 0)$.

2. 180° Rotation (Clockwise or Counterclockwise):

   • The rotation rule for 180° about the origin is $(x, y) \rightarrow (-x, -y)$.
   • $A(-2, 2)$ becomes $A'(2, -2)$.
   • $B(-3, 1)$ becomes $B'(3, -1)$.
   • $C(0, 0)$ remains $C'(0, 0)$.

3. 270° Counterclockwise Rotation (Equivalent to 90° Clockwise):

   • The rotation rule for 270° counterclockwise about the origin is $(x, y) \rightarrow (y, -x)$.
   • $A(-2, 2)$ becomes $A'(2, 2)$.
   • $B(-3, 1)$ becomes $B'(1, 3)$.
   • $C(0, 0)$ remains $C'(0, 0)$.

Explanation

• If you choose 90° counterclockwise, the triangle rotates in the positive direction (counterclockwise), resulting in new coordinates where each point’s x and y values are swapped and the y-coordinate is negated.
• For 180° rotation, both x and y coordinates of each point are negated, effectively flipping the triangle around the origin.
• For 270° counterclockwise (or 90° clockwise), the coordinates swap similarly to the 90° counterclockwise rotation, but the x-coordinate is negated instead of the y-coordinate.

Example of Writing About What You Did:

• "I rotated the pre-image triangle 90° counterclockwise about the origin. Using the rotation rule $(x, y) \rightarrow (-y, x)$, the vertices of the triangle moved as follows: point $A(-2, 2)$ rotated to $A'(-2, -2)$, point $B(-3, 1)$ rotated to $B'(-1, -3)$, and point $C(0, 0)$ remained unchanged."

Would you like to see more detailed steps or have any further questions?

Related Questions:

1. What is the effect of a 360° rotation on any point in a plane?
2. How does rotating a shape affect its orientation and size?
3. Can you describe the steps to rotate a shape 45° around a point that is not the origin?
4. How would the rotation rules change if the rotation center was $(1, 1)$ instead of $(0, 0)$?
5. How do you determine the center of rotation given an original shape and its image?

Tip:

When rotating points on a coordinate plane, always pay attention to the signs of the coordinates after applying the rotation rule, as this will determine the correct quadrant placement of the rotated image.