https://file.aiquickdraw.com/mathbot/file/1726752025820iyl6v73p.jpg

The image displays the problem involving triangle \( \Delta ABC \) with vertices \( A(7, -2) \), \( B(3, 9) \), and \( C(8, 6) \), where various rotations are applied to the points.

### 1. **Given Triangle Vertices**:
   - \( A(7, -2) \)
   - \( B(3, 9) \)
   - \( C(8, 6) \)

### 2. **Transformations Described**:

#### a) **90° Counterclockwise Rotation**:
   - The formula for a 90° counterclockwise rotation is:
     \[
     (x, y) \to (-y, x)
     \]
   - Applying this transformation:
     - \( A(7, -2) \to A'(-(-2), 7) = A'(2, 7) \)
     - \( B(3, 9) \to B'(-9, 3) = B'(-9, 3) \)
     - \( C(8, 6) \to C'(-6, 8) = C'(-6, 8) \)

   So, the rotated points are:
   - \( A'(2, 7) \)
   - \( B'(-9, 3) \)
   - \( C'(-6, 8) \)

#### b) **180° Counterclockwise Rotation**:
   - The formula for a 180° rotation is:
     \[
     (x, y) \to (-x, -y)
     \]
   - Applying this transformation:
     - \( A(7, -2) \to A'(-7, 2) \)
     - \( B(3, 9) \to B'(-3, -9) \)
     - \( C(8, 6) \to C'(-8, -6) \)

   The rotated points are:
   - \( A'(-7, 2) \)
   - \( B'(-3, -9) \)
   - \( C'(-8, -6) \)

These transformations have been correctly displayed on the board as examples of 90° and 180° counterclockwise rotations.

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Would you like more details on how to calculate these rotations or any further clarifications?

### Related Questions:
1. How do you determine the formula for a 270° counterclockwise rotation?
2. What is the effect of a 360° rotation on any point?
3. How do you derive the general rotation matrix for any angle \( \theta \)?
4. How does a reflection over the y-axis affect the coordinates of a point?
5. What are the differences between clockwise and counterclockwise rotations?

**Tip**: To visualize rotations, you can plot the points before and after rotation on a graph!

Given a triangle ABC with vertices A(7,-2), B(3,9), and C(8,6), apply 90° and 180° counterclockwise rotations to find the new coordinates of the vertices.

This problem involves applying 90° and 180° counterclockwise rotations to triangle ABC with vertices A(7,-2), B(3,9), and C(8,6). Learn how to transform points using key geometric rotation formulas.

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