2.4 Rigid Motions & Transformations

Rigid Motions: Preserving Size and Shape

A transformation maps every point of a figure (the pre-image) to a new point (the image). A rigid motion, also called an isometry, is a transformation that preserves distance and angle measure, so the image is the same size and shape as the pre-image. The three basic rigid motions are translations, reflections, and rotations. New York standards G-CO.A.2 through G-CO.A.5 cover describing these motions, and G-CO.B.6 ties them to congruence.

• A translation slides every point the same distance in the same direction (a "vector").
• A reflection flips a figure over a line of reflection; that line is the perpendicular bisector of every segment joining a point to its image, and points on the line stay fixed.
• A rotation turns a figure about a fixed center through a given angle and direction (counterclockwise is positive).

Because all three preserve length and angle, orientation of a reflection is reversed (a mirror image), while translations and rotations keep the original orientation.

What Rigid Motions Preserve

Knowing exactly what a rigid motion keeps unchanged lets you answer "which property is preserved" items instantly. Every rigid motion preserves distance (segment length), angle measure, betweenness of points, collinearity, and parallelism. As a result the image is congruent to the pre-image. What a rigid motion does not guarantee is orientation: a reflection reverses it. The mapping is written pre-image -> image, and every rigid motion has an inverse that sends each image point back to its pre-image, so the motions are reversible.

Coordinate Rules

The Regents rewards fast, accurate coordinate rules. Memorize this table (counterclockwise rotations are about the origin):

Worked examples. Point A(2, -1) translated 3 right and 5 down uses (x + 3, y - 5) -> (5, -6). Point P(-4, 7) reflected over the y-axis uses (-x, y) -> (4, 7). Point Q(3, -2) rotated 90° counterclockwise about the origin uses (-y, x) -> (2, 3). Point (2, -5) reflected over the line y = x uses (y, x) -> (-5, 2).

Symmetry

A figure has line symmetry (reflection symmetry) if a reflection over some line maps the figure exactly onto itself; that line is a line of symmetry. A square has 4 lines of symmetry, an equilateral triangle has 3, and a regular n-gon has n.

A figure has rotational symmetry if a rotation of less than 360° about its center maps it onto itself. The smallest such angle for a regular n-gon is 360° / n. For a regular octagon, that is 360 / 8 = 45° - a commonly tested value.

Composition of Transformations

A composition applies one transformation, then another, to the result. Apply them in order, using the image of the first step as the input to the second.

Worked example. Reflect P(-2, 3) over the y-axis, then translate 4 right and 1 down. First (-x, y) gives (2, 3); then (x + 4, y - 1) gives (6, 2). A glide reflection is a specific composition: a translation followed by a reflection over a line parallel to the translation.

Order can matter. A reflection over the x-axis followed by a reflection over the y-axis produces the same result as a single 180° rotation about the origin, so those two compositions are equal. But in general swapping the order of two different transformations changes the image, so always apply them in the sequence stated. A helpful fact: the composition of two reflections over parallel lines is a translation, and the composition of two reflections over intersecting lines is a rotation about the point where the lines meet.

Congruence Defined by Rigid Motions

The Next Generation standards define congruence through motion, not just matching measurements. Standard G-CO.B.6 states that two figures are congruent if and only if there is a sequence of rigid motions (translations, reflections, rotations) that maps one figure exactly onto the other. This is the modern, transformation-based definition you should cite in explanations.

Identifying a mapping sequence. Triangle ABC has vertices A(1, 2), B(4, 2), C(1, 5); its image has A'(-1, 3), B'(2, 3), C'(-1, 6). Compare a matched pair: A(1, 2) -> A'(-1, 3) shifts x by -2 and y by +1. Checking B and C shows the same shift, so a single translation 2 units left and 1 unit up maps ABC onto its image, proving the triangles congruent.

When one figure is a mirror image of the other, a reflection (or a rotation of 180°) must appear in the sequence; when orientation is preserved but the figure is turned, use a rotation. If any step changed the size, the map would be a dilation, not a rigid motion, and the figures would be similar rather than congruent - the distinction Chapter 4 develops.