Rotations

Geometry · Axiom Academy

Turning shapes around a center point — and the simple coordinate rules that describe every turn. A rotation turns a figure around a fixed center of rotation . Every point moves along a circular arc, staying the same distance from the center — so the figure never stretches or shrinks. Watch a single point swing counterclockwise around the origin: its distance from the center never changes. Positive angle Standard direction in math Negative angle Opposite of standard Every rotation needs a center point. In coordinate geometry we most often rotate around the origin (0, 0) . The center stays fixed while everything else turns around it. Key insight: the distance from any point to the center equals the distance from its image to the center. 2. Rotation Rules (About the Origin) Turning about the origin sends each point to a new point by a simple coordinate swap. Here are the three you use most. The animation rotates the point P(4, 2) counterclockwise and stops at each landmark angle so you can see the rule applied. 3. Visualizing 90° Counterclockwise Here is the 90° CCW rule built in the plane. The point P(4, 2) rides its dashed arc through a quarter-turn to its image P'(-2, 4) — same distance from the origin the whole way. Watch how the coordinates swap and one sign flips: . One table to keep handy. Notice that 90° clockwise and 270° counterclockwise land on the same rule — turning the short way one direction is the long way the other.

This is the written version of the interactive lesson above. See the full Geometry course.