Day 8: Rigid Transformations - Lesson Summary and Practice

Congruence and Rigid Transformations

• Congruent Figures: A figure is considered congruent to another if there exists a sequence of specific transformations—translations, rotations, and reflections—that maps one figure exactly onto the other.

  • Rigid Motions: Translations, rotations, and reflections are known as rigid motions. These transformations are fundamental because they preserve the size and shape of figures.

• Rigid Transformation: Any sequence of rigid motions is collectively termed a rigid transformation.

  • Key Property: A rigid transformation is defined by its characteristic of not changing measurements on any figure. This means that if polygons undergo a rigid transformation, their corresponding sides will maintain the same length, and their corresponding angles will retain the same measure.

• Terminology for Transformations:

  • Image: The result of any transformation is called the image.

  • Original Figure Points (Inputs): Points in the original figure, which serve as inputs for the transformation sequence, are typically named with capital letters (e.g., $A$, $B$, $C$).

  • Image Points (Outputs): Points in the image are the outputs and are named with capital letters followed by an apostrophe, referred to as "prime" (e.g., $A'$, $B'$, $C'$, read as "A prime, B prime, C prime"). For a second transformation in a sequence, double primes are used (e.g., $A''$, read as "A double prime").

• Sequences of Transformations:

  • Multiple sequences of transformations can lead to the same image, demonstrating congruence in various ways.

  • Order Matters: While some steps in a sequence might be switched without affecting the final output, altering the order of other steps can result in a different image. This emphasizes that sequences of transformations are not always commutative.

  • Congruence during Sequence: Each individual step in a sequence of rigid transformations (e.g., transforming $riangle ABC$ to $riangle A'B'C'$ and then to $riangle A''B''C''$) creates an intermediate figure that is congruent to the original figure.

Details on Translations

• Definition of Translation: A translation is a transformation that slides a figure in a specific direction for a given distance, entirely without any rotation.

• Specifying Direction and Distance: Both the distance and direction of a translation are precisely defined by a directed line segment.

  • Arrow: The arrow on the directed line segment indicates the specific direction of the translation.

  • Length: The length of the directed line segment specifies the distance or "how far" the figure is moved.

• Precise Definition for a Point: A translation of a point $A$ along a directed line segment $extbf{t}$ results in an image point $A'$ such that:

  1. The directed line segment $extbf{AA'}$ is parallel to $extbf{t}$.

  2. The directed line segment $extbf{AA'}$ points in the same direction as $extbf{t}$.

  3. The length of $extbf{AA'}$ is equal to the length of $extbf{t}$.

• Example (Points $C, D, E$ translated by $extbf{v}$): When three points $C, D, E$ are translated by a directed line segment $extbf{v}$, their image points $C', D', E'$ are formed. The individual directed line segments $extbf{CC'}$, $extbf{DD'}$, and $extbf{EE'}$ are all:

  • Parallel to $extbf{v}$

  • Going in the same direction as $extbf{v}$

  • Having the same length as $extbf{v}$

• Translations and Lines/Line Segments:

  • When line segments (e.g., $extbf{CE}$) are translated, their images (e.g., $extbf{C'E'}$) are parallel to the original segments.

  • Important Theorem: "Translations take lines to parallel lines or themselves." This means a line will either map to a different line that is parallel to the original, or if the translation vector is parallel to the line, it maps to itself (but all points shift along the line).

• Key Facts Related to Translations and Parallel Lines: These concepts are crucial for future lessons:

  1. Parallel Postulate: Given a line and a point not on that line, there exists one and only one parallel line that passes through the given point.

  2. Line Translation Property: Translations transform a line either into a parallel line or into itself. (Proof of this statement may be discussed in class).

Practice Problems & Concepts Illustrated

• 1. Identifying Non-Rigid Transformations: A rigid transformation preserves shape and size. Transformations that stretch, shrink, or distort a figure are not rigid transformations. For example, if a triangle's side lengths or angle measures change, the transformation is not rigid.

• 2. Correct Definition of Congruent: The correct definition of congruent figures is that there is a sequence of rigid transformations (rotations, reflections, and translations) that takes one figure exactly onto the other.

  • Incorrect definitions: Having the same shape (A) isn't sufficient as size must also be preserved. Having the same area (B) is also not sufficient, as figures with different shapes can have the same area. A generic sequence of transformations (C) is too broad, as it doesn't specify rigid transformations, which are critical for preserving congruence.

• 3. Applying Rigid Transformations: If a sequence of rigid transformations takes points $A$, $B$, and $C$ to $A'$, $B'$, and $C'$ respectively, then the same sequence applied to point $D$ will result in $D'$. The image point $D'$ will maintain its relative position and distance with respect to $A'B'C'$ as $D$ does to $ABC$. This indicates that the transformation maps the entire figure, preserving all geometric relationships.

• 4. Locating a Central Point for Multiple Schools (Equidistance): To locate a stadium roughly the same distance from three schools ($A, B, C$), one would look for the circumcenter of the triangle formed by the three school locations. The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect, and it is equidistant from all three vertices of the triangle.

• 5. Perpendicular Bisectors and Parallel Lines: If Hanford constructs the perpendicular bisector of line segment $AB$ and then draws line $CD$, there is no guarantee that line $CD$ will be parallel to line $AB$. Perpendicular bisectors are related to perpendicularity and equidistance, not directly to creating parallel lines in this manner. To construct a line parallel to $AB$ through point $C$, one would typically use alternate interior angles, corresponding angles, or a rigid translation.

• 6. Properties of a Perpendicular Line Construction: In a straightedge and compass construction of a line perpendicular to line $AB$ passing through point $C$, several statements must be true, often involving equal radii, perpendicularity, and segment bisection. These constructions rely on creating congruent triangles or specific geometric relationships (e.g., two points on the perpendicular bisector are equidistant from the endpoints of the segment).

• 7. Matching Translations to Directed Line Segments: This problem tests the understanding of how a directed line segment precisely dictates the direction and distance of a translation. Polygon $P$ is translated to Polygon $Q$ such that every point in $P$ moves along a path parallel to and the same length as the given directed line segment to reach its corresponding point in $Q$.

• 8. Drawing a Translated Image: To translate a quadrilateral $ABCD$ by a directed line segment $extbf{v}$, each vertex ($A, B, C, D$) must be shifted by the same direction and distance as specified by $extbf{v}$. The image points $A', B', C', D'$ will form a new quadrilateral congruent to the original, located relative to each original vertex by the vector $extbf{v}$.

• 9. True Statements about Translations:

  • (A) A translation takes a line to a parallel line or itself: This is a true statement, representing a fundamental property of translations.

  • (B) A translation takes a line to a perpendicular line: This is false. Translations preserve orientation; they do not rotate lines to be perpendicular.

  • (C) A translation requires a center of translation: This is false. A center of translation is not a concept for translations; a center of rotation is used for rotations.

  • (D) A translation requires a line of translation: This is false. A directed line segment (or vector) defines a translation, not a specific

• Congruent Figures & Rigid Transformations: Figures are congruent if a sequence of rigid motions (translations, rotations, reflections) maps one exactly onto the other. Rigid motions preserve the size and shape of figures, meaning corresponding sides maintain length and corresponding angles retain measure. The result of a transformation is an image (e.g., $A'$) from the original figure points (e.g., $A$). In sequences of transformations, order can matter, and each step produces an intermediate figure congruent to the original.

• Details on Translations: A translation slides a figure in a specific direction and distance, defined by a directed line segment. For a point $A$ translated by $extbf{t}$ to $A'$, the segment $extbf{AA'}$ is parallel to, in the same direction as, and equal in length to $extbf{t}$. Translations map lines to parallel lines or to themselves. This connects to the Parallel Postulate, stating there's one unique parallel line through a point not on a given line.

• Practice Concepts:

  • Non-Rigid Transformations: Change shape/size.

  • Congruent Definition: Requires a sequence of rigid transformations.

  • Applying Transformations: Maps entire figures, preserving geometric relationships.

  • Equidistance: Circumcenter is equidistant from three points.

  • Perpendicular Bisectors: Do not directly guarantee parallel lines.

  • Translations & Line Segments: Defined by a directed line segment, mapping a line to a parallel line or itself (e.g., "A translation takes a line to a parallel line or itself" is true).