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The problem requires rotating the triangle \( HFG \) 90 degrees counterclockwise about the point \( (3, 4) \).

### Steps to Rotate the Triangle:

1. **Identify the coordinates of points H, F, and G:**
   - From the image, let's determine the approximate coordinates:
     - \( H \) is at \( (3, 5) \)
     - \( F \) is at \( (4, 7) \)
     - \( G \) is at \( (6, 4) \)
   - The center of rotation is given as \( (3, 4) \).

2. **Translation to Origin:**
   - Subtract the center of rotation coordinates \( (3, 4) \) from each point to translate the points so the center of rotation becomes the origin:
     - \( H' = (3 - 3, 5 - 4) = (0, 1) \)
     - \( F' = (4 - 3, 7 - 4) = (1, 3) \)
     - \( G' = (6 - 3, 4 - 4) = (3, 0) \)

3. **Apply 90-Degree Counterclockwise Rotation:**
   - The formula for a 90-degree counterclockwise rotation around the origin is:
     \[
     (x', y') \to (-y', x')
     \]
   - Apply the formula to each translated point:
     - \( H'' = (0, 1) \to (-1, 0) \)
     - \( F'' = (1, 3) \to (-3, 1) \)
     - \( G'' = (3, 0) \to (0, 3) \)

4. **Translate Back to the Original Center:**
   - Add the coordinates of the original center of rotation \( (3, 4) \) to each rotated point:
     - \( H_{\text{new}} = (-1 + 3, 0 + 4) = (2, 4) \)
     - \( F_{\text{new}} = (-3 + 3, 1 + 4) = (0, 5) \)
     - \( G_{\text{new}} = (0 + 3, 3 + 4) = (3, 7) \)

### Final Rotated Coordinates:

- Point \( H \) is now at \( (2, 4) \).
- Point \( F \) is now at \( (0, 5) \).
- Point \( G \) is now at \( (3, 7) \).

### Conclusion:

The triangle \( HFG \), after a 90-degree counterclockwise rotation about the point \( (3, 4) \), has the new points:
- \( H = (2, 4) \)
- \( F = (0, 5) \)
- \( G = (3, 7) \)

Would you like more details or have any questions?

Here are 5 related questions you may find interesting:

1. How do we perform a rotation of any angle around a point that is not the origin?
2. How can we derive the general rotation formula for different degrees?
3. What happens to a shape when rotated 180 degrees around a point?
4. Can we find the new coordinates of any point after reflecting it over a specific line?
5. How can the rotation of shapes help in real-life applications, such as robotics or computer graphics?

**Tip:** Always remember to translate shapes to the origin for easier rotation calculations and then translate back to their original position.

Rotate the triangle 90 degrees counterclockwise about the point (3, 4).

Learn how to rotate the triangle HFG 90 degrees counterclockwise about the point (3, 4) with a detailed step-by-step solution. This problem involves geometric concepts like coordinate rotation and uses the 90-degree rotation formula. Suitable for students in Grades 8-10.

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