Geometric Transformations

Types of Transformations

Translation (Slide)

Every point moves the same distance in the same direction.

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

Translate right 3, up 2: $(x, y) \to (x + 3, y + 2)$

Reflection (Flip)

A mirror image across a line.

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

Rotation (Turn)

Turn around a center point (usually the origin).

| Rotation about origin | Rule | |---|---| | $90°$ counterclockwise | $(x, y) \to (-y, x)$ | | $180°$ | $(x, y) \to (-x, -y)$ | | $270°$ counterclockwise | $(x, y) \to (y, -x)$ |

Dilation (Resize)

Enlarge or shrink by a scale factor $k$ from a center point.

$(x, y) \to (kx, ky)$

• $k > 1$: enlargement
• $0 < k < 1$: reduction

Rigid Transformations (Congruence)

Translations, reflections, and rotations are rigid — they preserve:

• Size (lengths)
• Shape (angles)
• Congruence

Similarity Transformations

Dilations change size but preserve shape. Combined with rigid transformations, they create similar figures.

Congruence vs. Similarity

• Congruent: Same shape AND same size (rigid transformations)
• Similar: Same shape, possibly different size (dilation + rigid)

Coordinate practice: Triangle with vertices $(1,2), (3,4), (1,4)$. Reflect over the y-axis to get $(-1,2), (-3,4), (-1,4)$.