🔷 Geometry — Congruence
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Rigid Motions

Translate, rotate, and reflect figures while preserving shape and size.

A rigid motion (isometry) is a transformation that preserves both distance and angle measure. The three fundamental rigid motions are translations, rotations, and reflections. Compositions of rigid motions are also rigid motions.

Three Rigid Motions

Translation: slide every point by vector (a, b) → (x+a, y+b)Rotation: turn about a center point by angle θReflection: flip across a line of symmetry

Any two congruent figures are related by a composition of rigid motions.

Translation by (a,b):(x,y)↦(x+a,  y+b)\text{Translation by }(a,b): \quad (x, y) \mapsto (x+a,\; y+b)
Rotation by θ about origin:(x,y)↦(xcos⁡θ−ysin⁡θ,  xsin⁡θ+ycos⁡θ)\text{Rotation by }\theta\text{ about origin}: \quad (x,y) \mapsto (x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta)
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Two figures are congruent if and only if one can be mapped onto the other by a finite composition of rigid motions.

📐Reflecting a Point
Reflect point P(3, −2) across the y-axis. The rule for reflection across the y-axis is (x, y) → (−x, y). So P maps to (−3, −2).
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Remember This!

Common reflection rules: across x-axis → (x, y) → (x, −y); across y-axis → (x, y) → (−x, y); across y = x → (x, y) → (y, x).

✏️ Try It!

Point A(4, 2) is rotated 90° counterclockwise about the origin. What are the new coordinates?

Practice with CalcVerse
🔄Coordinate Transformations
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