Perform rotations

A rotation turns a figure about a specified center through a specified angle and direction, such as a quarter-turn, half-turn, or three-quarter turn; each point moves along a circular path while its distance from the center, side lengths, angle measures, and overall shape remain unchanged. Learners interpret and represent these rotations on grids or in the plane, distinguishing clockwise from counterclockwise motion and recognizing that the image may change position and orientation but not size; arbitrary-angle rotations and general coordinate rules are beyond this scope.

Detailed Explanation: Perform rotations

A rotation moves every point around the same center. The figure turns, but its size and shape stay the same.

The center of rotation stays fixed.
A quarter-turn is $90^\circ$ .
A half-turn is $180^\circ$ .
A three-quarter turn is $270^\circ$ .
Counterclockwise means turning left.
Clockwise means turning right.

To perform a rotation:

Locate the center of rotation.
Find the direction and angle of the turn.
For each point, keep the same distance from the center.
Move the point around the center through the given angle.
Connect the new points in the same order.

Example: Rotate triangle $ABC$ $90^\circ$ counterclockwise about the point $O(3,3)$ .

The vertices are

A(5,3), \qquad B(5,5), \qquad C(3,5).

Step 1: Rotate point $A$

Point $A$ is $2$ units to the right of $O$ . A $90^\circ$ counterclockwise turn moves it $2$ units above $O$ .

So,

A(5,3)\longrightarrow A'(3,5).

Step 2: Rotate point $B$

Point $B$ is $2$ units right and $2$ units above $O$ . After a $90^\circ$ counterclockwise turn, it is $2$ units left and $2$ units above $O$ .

So,

B(5,5)\longrightarrow B'(1,5).

Step 3: Rotate point $C$

Point $C$ is $2$ units above $O$ . After a $90^\circ$ counterclockwise turn, it is $2$ units left of $O$ .

So,

C(3,5)\longrightarrow C'(1,3).

Step 4: Draw the image

Plot $A'(3,5)$ , $B'(1,5)$ , and $C'(1,3)$ , then connect them.

The rotated triangle is $A'B'C'$ . It has the same side lengths and angles as the original triangle, but its position and orientation have changed.

Learn by doing: Perform rotations

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Cartesian Grid - Rotation of Point (Grid to Grid) around Point on Axis

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Perform rotations

Detailed Explanation: Perform rotations

Step 1: Rotate point AAA

Step 2: Rotate point BBB

Step 3: Rotate point CCC