Skill: Apply coordinate rules for rotations

A rotation moves every point the same amount around the origin. For common rotations, use these coordinate rules:

• $90^\circ$ counterclockwise: $(x,y)\rightarrow(-y,x)$
• $180^\circ$: $(x,y)\rightarrow(-x,-y)$
• $270^\circ$ counterclockwise: $(x,y)\rightarrow(y,-x)$

For a clockwise rotation, use the rule for the opposite direction. For example, a $90^\circ$ clockwise rotation uses $(x,y)\rightarrow(y,-x)$.

Example

Rotate triangle $ABC$ $90^\circ$ counterclockwise about the origin.

$$A(2,1),\qquad B(4,1),\qquad C(2,3)$$

For a $90^\circ$ counterclockwise rotation, replace $(x,y)$ with $(-y,x)$.

Apply the rule to each vertex:

• Point $A(2,1)$:

$$A'(2,1)\rightarrow(-1,2)$$

• Point $B(4,1)$:

$$B'(4,1)\rightarrow(-1,4)$$

• Point $C(2,3)$:

$$C'(2,3)\rightarrow(-3,2)$$

Therefore, the rotated triangle has vertices

$$\boxed{A'(-1,2),\quad B'(-1,4),\quad C'(-3,2)}$$

Be careful to change the signs and switch the order of the coordinates exactly as the rule shows. A rotation changes the location of a figure but preserves its side lengths and angle measures.