Skill: Rotate figures about a point

A rotation turns a figure around a fixed point called the center of rotation. Every point moves the same angle in the same direction. The figure keeps its size and shape.

For rotations about the origin:

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

The origin stays in the same place because it is the center of rotation.

Example

Rotate triangle $ABC$ with vertices

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

$90^\circ$ counterclockwise about the origin.

Step 1: Use the rotation rule

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

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

This means:

1. Change the sign of the $y$-coordinate.
2. Put that new value first.
3. Use the original $x$-coordinate as the new $y$-coordinate.

Step 2: Rotate each vertex

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

Step 3: Draw the rotated figure

Plot the new points:

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

Connect $A'$ to $B'$, $B'$ to $C'$, and $C'$ to $A'$. The result is triangle $A'B'C'$, which is the original triangle turned $90^\circ$ counterclockwise around the origin.

Always apply the rotation rule to every vertex, then connect the new points in the same order.