To rotate each of the points $Z(-1, -5)$, $K(-1, 0)$, $C(1, 1)$, and $N(3, -2)$ by $270^\circ$ clockwise about the origin, we’ll use the rule for a $270^\circ$ clockwise rotation. For any point $(x, y)$, rotating it $270^\circ$ clockwise about the origin transforms it to the point $(y, -x)$.

Let’s apply this transformation to each point:

1. Point $Z(-1, -5)$:

   • After rotation: $(-5, 1)$

2. Point $K(-1, 0)$:

   • After rotation: $(0, 1)$

3. Point $C(1, 1)$:

   • After rotation: $(1, -1)$

4. Point $N(3, -2)$:

   • After rotation: $(-2, -3)$

Final Coordinates After $270^\circ$ Clockwise Rotation:

• $Z' = (-5, 1)$
• $K' = (0, 1)$
• $C' = (1, -1)$
• $N' = (-2, -3)$

Would you like a step-by-step explanation of the transformation process, or is there anything specific you want to clarify?

Additional Related Questions:

1. How does a $270^\circ$ clockwise rotation differ from a $90^\circ$ counterclockwise rotation?
2. How would the coordinates change if we used a different center of rotation?
3. What is the effect of rotating these points $180^\circ$ clockwise instead?
4. Could you calculate the distance between each original and rotated point?
5. How can we apply matrix multiplication to perform this rotation?

Tip:

For any $270^\circ$ clockwise rotation, simply switch $x$ and $y$, then make $x$ negative for the new coordinates.