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!