Sequences of Rigid Transformations
Sequences of rigid transformations is a Grade 7 geometry concept in Big Ideas Math Advanced 2, Chapter 2: Transformations. When multiple rigid motions — translations, rotations, and reflections — are applied in sequence, each transformation is applied to the result of the previous one, with prime notation tracking each step. The final image is always congruent to the original figure because all rigid motions preserve size and shape.
Key Concepts
Property A sequence of transformations is a series of transformations applied one after another to a figure. When you combine multiple rigid motions (like a rotation followed by a translation), the final image always maintains perfect congruence with the original starting figure.
Examples Two Steps: Triangle $ABC$ is first rotated $90^\circ$ counterclockwise about the origin, then translated $3$ units right to create triangle $A''B''C''$. Three Steps: Pentagon $DEFGH$ is translated $2$ units up, then rotated $270^\circ$ clockwise about point $D'$, producing pentagon $D''E''F''G''H''$.
Explanation When performing sequences, order matters, and you must apply each new move to the result of the previous move. To keep from getting lost, we use "prime notation" to track our steps. The original figure is plain ($A$). After step one, it gets a single prime ($A'$). After step two, it gets a double prime ($A''$). No matter how many prime marks a shape collects, if you only used rigid motions, the final shape is still $100\%$ congruent to the original.
Common Questions
What is a sequence of rigid transformations?
A sequence applies two or more rigid motions (translations, rotations, reflections) one after another. The image from each transformation becomes the pre-image for the next.
How do you track vertices through a sequence of transformations?
Use prime notation: the original is A, after the first transformation it is A prime, after the second it is A double prime, and so on. Apply each transformation to the most recent prime version.
Is the final figure congruent to the original in a sequence of rigid transformations?
Yes. Since translations, rotations, and reflections are all rigid motions that preserve size and shape, any sequence of them always results in a figure congruent to the original.
What textbook covers sequences of rigid transformations in Grade 7?
Big Ideas Math Advanced 2, Chapter 2: Transformations covers sequences of rigid transformations using prime notation to track intermediate images.