https://file.aiquickdraw.com/mathbot/file/1730255339742amd4ytkc.jpg

The problem involves finding the vertices of the image \( T^{(90^\circ, O)}(\triangle BCD) \), which means we need to rotate the triangle \( \triangle BCD \) 90 degrees counterclockwise around the origin (O). 

Let's go through the steps to determine the new coordinates for each vertex after the rotation:

1. **Original Coordinates of Points B, C, and D**:
   - \( B \): \((-3, 3)\)
   - \( C \): \( (3, 1)\)
   - \( D \): \((-2, -3)\)

2. **Applying a 90° Counterclockwise Rotation**:
   - For a 90° counterclockwise rotation around the origin, the rule is:  
     \((x, y) \rightarrow (-y, x)\)

3. **Calculate the New Coordinates**:
   - **B'**: For \( B(-3, 3) \), applying the rotation gives:
     \[
     B' = (-3, -3)
     \]
   - **C'**: For \( C(3, 1) \), applying the rotation gives:
     \[
     C' = (-1, 3)
     \]
   - **D'**: For \( D(-2, -3) \), applying the rotation gives:
     \[
     D' = (3, -2)
     \]

4. **Answer**:
   - \( B' = (-3, -3) \)
   - \( C' = (-1, 3) \)
   - \( D' = (3, -2) \)

Would you like further details or explanations on the rotation transformation?

### Follow-up Questions
1. What is the general formula for rotating a point 180 degrees about the origin?
2. How does a 270° rotation about the origin affect the coordinates of a point?
3. Can the rule \((x, y) \rightarrow (-y, x)\) be applied to rotations in the clockwise direction?
4. How do rotations affect the lengths of sides in a triangle?
5. What are the coordinates of point \( B \) if the rotation angle was 45° instead?

### Tip
For 90° rotations, always remember the rule: \((x, y) \rightarrow (-y, x)\) for counterclockwise and \((x, y) \rightarrow (y, -x)\) for clockwise rotation.

What are the vertices of the image T(90°, O)(△BCD)?

Explore how to find the vertices of △BCD after a 90° counterclockwise rotation around the origin. This example demonstrates key geometry concepts, including coordinate transformations and rotation rules, suitable for Grades 8-10.

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