Transformations

1. Introduction

A transformation is a rule that moves every point of a figure to a new location, producing an image of the original (the pre-image). Transformations are how geometry captures motion and symmetry: sliding a chess piece, flipping a photo, spinning a steering wheel, or zooming a map are all transformations. They also unify the rest of geometry — congruence is "related by rigid motions" and similarity is "related by a dilation plus rigid motions."

There are four core transformations in the plane. Three of them — translations, reflections, and rotations — are rigid motions (isometries): they preserve every distance and angle, so the image is congruent to the pre-image. The fourth, dilation, stretches or shrinks a figure by a scale factor, preserving shape but not size, so the image is similar (and congruent only when the scale factor is $1$).

Working in the coordinate plane, each transformation becomes a precise algebraic rule on coordinates $(x, y)$. This article develops each rule, explains why it works, shows how to compose transformations, studies symmetry, and extends rotations and dilations to centers other than the origin. Sketch the points as you read — plotting a pre-image and image side by side makes every rule obvious.

Transformations are also the rigorous definition of congruence and similarity. Saying two triangles are congruent is equivalent to saying one can be carried to the other by a sequence of translations, rotations, and reflections. Saying they are similar adds a dilation to that list. The coordinate rules you memorize are the computational face of these geometric ideas.

2. Core Concepts

2.1 Pre-image, Image, and Notation

The starting figure is the pre-image; the result is the image. We label image points with primes: point $A$ maps to $A'$ (read "A prime"), and after a second transformation to $A''$. A transformation rule is written as a mapping, e.g. $(x, y) \to (x + 3, y - 2)$, meaning "replace each point by the one $3$ right and $2$ down."

2.2 Translations (Slides)

A translation shifts every point by the same horizontal amount $a$ and vertical amount $b$: $(x, y) \to (x + a,\, y + b).$ Every point moves the same distance in the same direction, so the figure neither turns nor flips. A translation is a rigid motion; the image is congruent and has the same orientation. The vector $\langle a, b\rangle$ fully describes the slide.

2.3 Reflections (Flips)

A reflection flips a figure across a line of reflection, as if that line were a mirror. Each point and its image are the same distance from the line, on opposite sides, and the line is the perpendicular bisector of the segment joining a point to its image. Reflections reverse orientation (a clockwise lettering becomes counterclockwise). The standard coordinate rules:

• Across the $x$-axis: $(x, y) \to (x, -y)$ (negate $y$).
• Across the $y$-axis: $(x, y) \to (-x, y)$ (negate $x$).
• Across the line $y = x$: $(x, y) \to (y, x)$ (swap coordinates).
• Across the line $y = -x$: $(x, y) \to (-y, -x)$.

A point on the mirror line stays fixed (it is its own image).

2.4 Rotations (Turns)

A rotation turns a figure about a fixed center through a directed angle. By convention, counterclockwise (CCW) is positive. Distances from the center are preserved, so the image is congruent and orientation is unchanged. The most common rotations about the origin:

• $90^\circ$ CCW: $(x, y) \to (-y, x)$.
• $180^\circ$: $(x, y) \to (-x, -y)$.
• $270^\circ$ CCW (same as $90^\circ$ clockwise): $(x, y) \to (y, -x)$.

Where does the $90^\circ$ rule come from? Rotating a point $90^\circ$ CCW about the origin keeps its distance to the origin the same and turns its direction a quarter turn; in coordinates this exactly swaps and sign-adjusts the entries to $(-y, x)$. You can verify with a simple point: $(1, 0)$ (pointing right) rotates to $(0, 1)$ (pointing up), and indeed $(-0, 1) = (0, 1)$.

2.5 Dilations (Resizing)

A dilation with center $C$ and scale factor $k$ moves each point along the ray from $C$ so its distance from $C$ is multiplied by $k$. Centered at the origin: $(x, y) \to (kx,\, ky).$ If $|k| > 1$ the figure enlarges; if $0 < |k| < 1$ it shrinks; a negative $k$ also flips it through the center. A dilation preserves angles and shape (so the image is similar) but multiplies all lengths by $|k|$, areas by $k^2$. It is the only one of the four that is not a rigid motion (unless $k = \pm 1$).

2.6 Compositions and Symmetry

A composition applies one transformation, then another; apply them in order, feeding each image into the next rule. Two reflections over parallel lines compose to a translation; two reflections over intersecting lines compose to a rotation about the intersection point. A figure has line symmetry if a reflection maps it onto itself, and rotational symmetry if a rotation (less than a full turn) does. These ideas let you classify the symmetry of polygons and tilings.

2.7 Rotation About a Point Other Than the Origin

To rotate point $P$ by $\theta$ about center $C(h, k)$: translate so $C$ moves to the origin ($(x,y) \to (x-h, y-k)$), apply the origin rotation rule, then translate back ($(x,y) \to (x+h, y+k)$). For $90^\circ$ CCW about $(h,k)$: $(x,y) \to (h - (y-k),\, k + (x-h))$.

2.8 Dilation About a Point Other Than the Origin

To dilate about center $C(h,k)$ with factor $k$: $(x,y) \to (h + k(x-h),\, k + k(y-k))$. Each coordinate moves $k$ times as far from $C$ as before. The origin rule $(kx, ky)$ is the special case $h = k = 0$.

2.9 Orientation and Rigid Motions

Orientation means the cyclic order of vertices (clockwise vs. counterclockwise). Translations and rotations preserve orientation; reflections reverse it. A composition of two reflections over intersecting lines is a rotation (orientation preserved); two reflections over parallel lines give a translation (also preserved). An odd number of reflections reverses orientation; an even number preserves it.

2.10 Symmetry in Regular Polygons

A regular $n$-gon has $n$ lines of reflection symmetry (through midpoints of sides and through vertices) and rotational symmetry of order $n$ (rotate by $\tfrac{360^\circ}{n}$). A square has $4$ rotational symmetries ($90^\circ$, $180^\circ$, $270^\circ$, $360^\circ$) and $4$ reflection lines. Recognizing symmetry shortcuts many transformation problems.

2.11 Glide Reflections (Preview)

A glide reflection combines a reflection across a line with a translation parallel to that line. It is an indirect isometry (reverses orientation) and appears in wallpaper patterns. Two parallel reflections with a translation between them produce this motion.

2.12 Transformations and Congruence/Similarity

Two figures are congruent iff a sequence of rigid motions maps one to the other.

2.13 Finding the Line of Reflection from Coordinates

If you know a point $(x, y)$ and its image $(x', y')$ under a reflection, the mirror line is the perpendicular bisector of the segment joining them. For reflections across the coordinate axes or $y = \pm x$, the rules are fixed. For arbitrary lines, the midpoint $\left(\tfrac{x+x'}{2}, \tfrac{y+y'}{2}\right)$ lies on the mirror, and the slope of the mirror is the negative reciprocal of the slope from $(x,y)$ to $(x',y')$. They are similar iff a dilation followed by rigid motions does the job. This formal definition connects the coordinate rules you use in class to the geometric meaning of "same size and shape" versus "same shape, different size."