Applying Motions to Problem Solving
Big Idea: Geometric motions (transformations) are powerful tools that can simplify complex geometry problems. By translating, rotating, or reflecting figures, we can reveal hidden relationships, prove congruence, and find elegant solutions.
1. Review of the Three Rigid Motions
Rigid motions (also called isometries) preserve distances and angle measures. The three fundamental rigid motions are:
Translation
A slide that moves every point the same distance in the same direction.
Notation: $T_{\langle a, b \rangle}(x, y) = (x + a, y + b)$
A translation is defined by a vector $\langle a, b \rangle$.
Rotation
A turn about a fixed center point by a given angle.
Notation: $R_{(c, \theta)}$ rotates by angle $\theta$ about center $c$.
For rotation about the origin by angle $\theta$:
$(x,y) \to (x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta)$
Reflection
A flip over a line (the line of reflection).
Common reflections:
- Over $x$-axis: $(x,y) \to (x, -y)$
- Over $y$-axis: $(x,y) \to (-x, y)$
- Over $y = x$: $(x,y) \to (y, x)$