Rigid Transformations

What Are Rigid Transformations?

Transformations that preserve size and shape (distances and angles stay the same). Also called isometries.

Translations (Slides)

Every point moves the same distance in the same direction.

$(x, y) \to (x + a, y + b)$

Example: Translate by $\langle 3, -2 \rangle$: $(1, 4) \to (4, 2)$

Reflections (Flips)

Every point is mirrored across a line of reflection.

| Reflection Over | Rule | |-----------------|------| | x-axis | $(x, y) \to (x, -y)$ | | y-axis | $(x, y) \to (-x, y)$ | | $y = x$ | $(x, y) \to (y, x)$ | | $y = -x$ | $(x, y) \to (-y, -x)$ |

Rotations (Turns)

Rotating about the origin counterclockwise:

| Angle | Rule | |-------|------| | $90°$ | $(x, y) \to (-y, x)$ | | $180°$ | $(x, y) \to (-x, -y)$ | | $270°$ | $(x, y) \to (y, -x)$ |

Congruence Through Transformations

Two figures are congruent if one can be mapped to the other using a sequence of rigid transformations.

$\triangle ABC \cong \triangle DEF$

means there exists a combination of translations, reflections, and/or rotations that maps $\triangle ABC$ exactly onto $\triangle DEF$.

Key principle: Rigid transformations preserve distances, angle measures, and parallelism.