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To solve this problem, we need to rotate the point \( (4, 5) \) by 90 degrees counter-clockwise about the point \( (2, 2) \).

### Steps:
1. **Translate the points to the origin:**
   - Subtract the point of rotation \( (2, 2) \) from the point \( (4, 5) \) to get:
     \[
     (4 - 2, 5 - 2) = (2, 3)
     \]
   Now, we are working with the point \( (2, 3) \) at the origin.

2. **Apply the 90-degree counter-clockwise rotation:**
   - The formula for a 90-degree counter-clockwise rotation is:
     \[
     (x', y') = (-y, x)
     \]
   So, applying this to the point \( (2, 3) \):
     \[
     (x', y') = (-3, 2)
     \]

3. **Translate the points back:**
   - Add the point of rotation \( (2, 2) \) back to the rotated coordinates \( (-3, 2) \):
     \[
     (-3 + 2, 2 + 2) = (-1, 4)
     \]

### Final Answer:
The new coordinates after the 90-degree counter-clockwise rotation are \( (-1, 4) \).

Would you like further details on this? Here are 5 related questions to explore this topic:

1. How do you rotate a point by 180 degrees about a given point?
2. What is the formula for rotating a point 90 degrees clockwise?
3. How would you rotate a point 45 degrees about the origin?
4. Can you explain how translation and rotation work together in geometry?
5. What are the effects of rotation on the distance between points?

**Tip:** When rotating points, always ensure to translate to the origin before applying the rotation formula for accurate results!

Rotate (4, 5) by 90 degrees counter-clockwise about the point (2, 2). What are the new coordinates?

Learn how to rotate the point (4, 5) by 90 degrees counter-clockwise about the point (2, 2). This problem covers key geometric concepts such as rotations and translations, with a detailed step-by-step solution. Suitable for students in Grades 8-10.

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