Rotations on the Plane

In geometry, a rotation is a transformation that turns a figure around a fixed point called the center of rotation. For Grade 8 math, we almost always use the origin $(0,0)$ as our pivot point. Imagine pinning a shape to the center of a graph with a thumbstack and spinning it. The shape stays the same size and shape (it is congruent), but its orientation changes. We measure these turns in degrees: $90^\circ$ (a quarter turn), $180^\circ$ (a half turn), and $270^\circ$ (a three-quarter turn). Unless specified otherwise, we always rotate counter-clockwise.

Instead of guessing where a shape lands, we use specific rules for the coordinates $(x, y)$. These rules are like 'functions' for movement. For a **$90^\circ$ counter-clockwise** rotation, the rule is $(x, y) \rightarrow (-y, x)$. For a **$180^\circ$** rotation, the rule is $(x, y) \rightarrow (-x, -y)$—essentially, both signs just flip! Finally, for a **$270^\circ$ counter-clockwise** (which is the same as $90^\circ$ clockwise), the rule is $(x, y) \rightarrow (y, -x)$. Memorizing these three patterns allows you to rotate complex polygons in seconds.

A figure has rotational symmetry if it can be rotated less than $360^\circ$ around its center and still look exactly like the original. The order of symmetry is the number of times the shape matches itself during a full $360^\circ$ turn. For example, a square has an order of 4 because it looks the same at $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$. An equilateral triangle has an order of 3. This concept is vital in architecture, nature (like starfish), and logo design.