Transformations

Translations, reflections, rotations, and dilations — in coordinates.

The idea

A transformation of the plane sends every point to a new point, and so carries any figure to an image. In coordinates, a transformation is a rule that turns the pair $(x, y)$ into another pair; to transform a figure, apply the rule to each of its points.

Four transformations are fundamental. A translation slides every point by the same vector, adding a fixed amount to each coordinate. A reflection flips the plane across a mirror line; a point and its image lie at equal distances on opposite sides of the line. A rotation turns the plane about a fixed center through a fixed angle. A dilation with center $O$ and scale factor $k$ moves each point along the ray from $O$ to $k$ times its distance from $O$.

The first three are rigid motions: they preserve distances, so the image is congruent to the original. A dilation multiplies every distance by $k$, so its image is similar to the original rather than congruent. All four preserve angles.

The figure shows one shape and its image under each of the four transformations.

Ways to work on it

• Walkthrough. Each transformation as a rule on coordinates.
• Practice. Apply a transformation to a point.
• Hardest. Rotations and diagonal reflections by their coordinate rule.

Not sure where to start? Take the ten-question placement test.