Skill: Rotate figures about the origin

A rotation about the origin moves every point the same amount around $(0,0)$. The figure keeps the same size and shape.

For quarter-turns, use these coordinate rules:

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

A $90^\circ$ clockwise turn is the same as a $270^\circ$ counterclockwise turn.

Example

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

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

Because the rotation is $90^\circ$ counterclockwise, use

$$(x,y)\to(-y,x).$$

Apply the rule to each vertex:

For $A(1,2)$:

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

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

For $B(4,2)$:

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

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

For $C(1,5)$:

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

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

The rotated triangle has vertices

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

To rotate a whole figure, apply the same coordinate rule to every vertex, then connect the new points in the same order.