Rotations and Reflections of Shapes

Core Idea

A reflection flips a shape over a line (like a mirror). A rotation turns a shape around a point. Both transformations preserve the shape and size. Tracing and folding activities make these concrete.

Explainer

You have already explored line symmetry — the idea that some shapes can be folded along a line so that both halves match perfectly. A reflection is exactly that fold, applied as a transformation. When you reflect a shape over a line, you get a mirror image on the other side. Every point of the shape travels straight across the line of reflection and lands the same distance on the other side. The line of reflection acts like a mirror, and the resulting shape is a perfect flip of the original.

A rotation is different: instead of flipping, you turn. Pick a point — called the center of rotation — and spin the shape around it, like the hand of a clock spinning around its center pin. The shape can rotate a quarter turn (90°), a half turn (180°), or any other amount. After rotating, the shape looks like the original tilted at an angle. A square rotated 90° around its center looks the same as before because of its symmetry — but a triangle rotated 90° will appear to be lying on its side.

Here is what makes both transformations special: the shape and size are preserved. A reflected triangle is still a triangle with the same side lengths and angles. A rotated hexagon still has six equal sides. Mathematicians call transformations that preserve shape and size rigid motions or isometries — the figure moves without stretching or shrinking. This is different from scaling (making a shape bigger or smaller), which changes size.

A practical way to tell a reflection from a rotation: after a reflection, the shape appears flipped as if seen in a mirror — letters like "b" become "d." After a rotation, the shape is tilted but not flipped — "b" might appear as "q" (rotated 180°) but nothing is reversed. If you trace a shape on tracing paper, you can physically perform both transformations: fold the paper to reflect, or spin it to rotate.