To solve this problem, we need to find the image of the point $(5, -9)$ after a rotation of $270^\circ$ counterclockwise about the origin.

Steps:

1. Understand the rotation rule:

   • A $270^\circ$ counterclockwise rotation is equivalent to a $90^\circ$ clockwise rotation. The transformation rule for $270^\circ$ counterclockwise is: $(x, y) \to (y, -x)$

2. Apply the transformation:

   • The original point is $(5, -9)$.
   • Using the rule: $(x, y) \to (y, -x) = (-9, -5)$

3. Result:

   • The image of the point $(5, -9)$ after a $270^\circ$ counterclockwise rotation is: $(-9, -5)$

Would you like me to provide more detailed explanations about the steps or rotation rules? Let me know!

Related Questions:

1. What is the image of the same point under a $90^\circ$ counterclockwise rotation?
2. How would the result change for a $180^\circ$ rotation about the origin?
3. Can we visualize this transformation on a coordinate plane?
4. What is the matrix representation of a $270^\circ$ counterclockwise rotation?
5. How does this transformation affect the quadrant location of the point?

Tip:

For rotations about the origin, always remember the specific transformation rules for $90^\circ$, $180^\circ$, and $270^\circ$. These can save time during calculations!