Skill: Rotate figures about the origin

To rotate a figure about the origin, apply the rotation rule to every vertex. The origin is the point $(0,0)$, and the figure keeps the same size and shape.

For a $90^\circ$ counterclockwise rotation, use:

$$(x,y)\rightarrow(-y,x)$$

This means:

1. Change the order of the coordinates.
2. Make the new first coordinate the opposite of the old $y$-coordinate.
3. Keep the old $x$-coordinate as the new $y$-coordinate.

Example

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

$$A(1,2),\qquad B(4,2),\qquad C(1,5)$$

Apply $(x,y)\rightarrow(-y,x)$ to each point.

For $A(1,2)$:

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

So, $A'(-2,1)$.

For $B(4,2)$:

$$(4,2)\rightarrow(-2,4)$$

So, $B'(-2,4)$.

For $C(1,5)$:

$$(1,5)\rightarrow(-5,1)$$

So, $C'(-5,1)$.

The rotated triangle has vertices:

$$\boxed{A'(-2,1),\ B'(-2,4),\ C'(-5,1)}$$

Remember the common quarter- and half-turn rules:

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