Rotate figures about a point

A rotation turns every point of a figure through the same angle and direction around a fixed center, leaving the center unchanged and preserving lengths, angle measures, and overall shape. In the coordinate plane, learners interpret and apply quarter-turn and half-turn rotations—especially $90^\circ$ , $180^\circ$ , and $270^\circ$ about the origin—using diagrams and coordinate rules, distinguishing clockwise from counterclockwise motion; arbitrary-angle formulas and more advanced rotation methods are beyond this scope.

Detailed Explanation: 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:

Change the sign of the $y$ -coordinate.
Put that new value first.
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.

Learn by doing: Rotate figures about a point

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Cartesian Grid - Rotation of Point (Coordinates to Coordinates) around Origin

Earned ?