Coordinate Rules for Rotations (About the Origin)

Property When the center of rotation is the origin $(0,0)$, we use algebraic rules to find the new coordinates. Assume all angles are Counterclockwise (CCW) unless stated otherwise. $90^\circ$ CCW: $(x, y) \rightarrow ( y, x)$ [Swap the numbers, change the sign of the NEW first number] $180^\circ$: $(x, y) \rightarrow ( x, y)$ [Do NOT swap numbers, just change BOTH signs] $270^\circ$ CCW (or $90^\circ$ CW): $(x, y) \rightarrow (y, x)$ [Swap the numbers, change the sign of the NEW second number].

Examples $90^\circ$ CCW Rotation: Rotate $A(4, 5)$. Step 1 (Swap): $(5, 4)$ Step 2 (Change first sign): $( 5, 4)$. So, $A'( 5, 4)$. $180^\circ$ Rotation: Rotate $B( 2, 7)$. Keep the order, flip both signs: $(+2, 7)$. So, $B'(2, 7)$. $90^\circ$ CW (which is $270^\circ$ CCW): Rotate $C( 3, 6)$. Step 1 (Swap): $( 6, 3)$ Step 2 (Change second sign): $( 6, +3)$. So, $C'( 6, 3)$.

Explanation This is where the most errors happen! Students often try to swap the numbers and change the signs at the exact same time in their heads, which leads to messy mistakes with negatives. Always do it in two micro steps: Write down the swapped numbers first, then apply the negative sign to the correct position. Also, remember that a $180^\circ$ rotation rule $( x, y)$ looks exactly like reflecting across both the x and y axes!