Grades Grade 8 GeometryTransformations: Translations, Rotations, Reflections & Dilations

Transformations: Translations, Rotations, Reflections & Dilations

📐 Geometry

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Transformations: Translations, Rotations, Reflections & Dilations

Rigid motions preserve shape and size; dilations preserve shape but change size.

A transformation moves or resizes a figure in the plane. Translations, rotations, and reflections are rigid motions (isometries) — they preserve distance and angle measures, so the image is congruent to the original. Dilations scale the figure by a factor k, producing a similar (but not necessarily congruent) image.

The four transformations:

Translation: slide every point (x, y) → (x + a, y + b)Rotation: turn around a center point by a given angle (90°, 180°, 270°)Reflection: flip over a line (x-axis, y-axis, y = x, etc.)Dilation: scale from a center point by factor k — (x, y) → (kx, ky)

\text{Translation by } (a, b): \; (x, y) \rightarrow (x+a, \, y+b)

\text{Dilation by factor } k \text{ about origin}: \; (x, y) \rightarrow (kx, \, ky)

🔺Reflection over the x-axis

Triangle with vertices A(1, 3), B(4, 3), C(2, 6).

Reflect over the x-axis: (x, y) → (x, −y)
A'(1, −3), B'(4, −3), C'(2, −6)

The shape is congruent (same size and shape, flipped).

Rigid Motions → Congruent

TranslationRotationReflectionAll distances and angles preserved

Dilation → Similar

Scale factor k enlarges or shrinksAngles preserved, lengths scaled by kIf k = 1, the image is congruentIf k ≠ 1, image is similar but not congruent

Congruent figures are identical in size and shape; similar figures have the same shape but proportional sizes.

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Remember This!

Rotation 90° counterclockwise: (x, y) → (−y, x). Rotation 180°: (x, y) → (−x, −y). Memorize these for quick coordinate transformations.

✏️ Try It!

Point P is at (3, −5). After a translation of (−4, 2), where is P'?

Practice with CalcVerse

🔄Coordinate Transformations

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