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10.1: Transformations Using Rigid Motions

  • Page ID
    32000
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    In this section we will learn about isometry or rigid motions. An isometry is a transformation that preserves the distances between the vertices of a shape. A rigid motion does not affect the overall shape of an object but moves an object from a starting location to an ending location. The resultant figure is congruent to the original figure.

    A rigid motion is when an object is moved from one location to another and the size and shape of the object have not changed.
    Two figures are congruent if and only if there exists a rigid motion that sets up a correspondence of one figure as the image of the other. Side lengths remain the same and interior angles remain the same.
    An identity motion is a rigid motion that moves an object from its starting location to exactly the same location. It is as if the object has not moved at all.

    There are four kinds of rigid motions: translations, rotations, reflections, and glide-reflections. When describing a rigid motion, we will use points like P and Q, located on the geometric shape, and identify their new location on the moved geometric shape by P' and Q'.

    We will start with the rigid motion called a translation. When translating an object, we move the object in a specific direction for a specific length, along a vector .

    Figure \(\PageIndex{1}\): Translation

    The translation of the blue triangle with point P was moved along the vector to the location of the red triangle with point P'. Also note that the other vertices of the blue triangle also moved along the vector to corresponding vertices on the red triangle.

    P'

    P


    A translation of an object moves the object along a directed line segment called a vector for a specific distance and in a specific direction. The motion is completely determined by two points P and P' where P is on the original object and P' is on the translated object.

    In regular language, a translation of an object is a slide from one position to another. You are given a geometric figure and an arrow which represents the vector. The vector gives you the direction and distance which you slide the figure.

    Example \(\PageIndex{1}\) Translation of a Triangle

    You are given a blue triangle and a vector . Move the triangle along vector .

    Figure \(\PageIndex{2}\): Blue Triangle and Vector

    B

    A C

    Figure \(\PageIndex{3}\): Result of the Translation

    B’


    A’ C’
    B


    A C

    Properties of a Translation

    1. A translation is completely determined by two points P and P’
    2. Has no fixed points
    3. Has identity motion

    Note: the vector has the same length as vector , but points in the opposite direction.

    Example \(\PageIndex{2}\) Translation of an Object

    Given the L-shape figure below, translate the figure along the vector . The vector moves horizontally three units to the right and vertically two units up. Move each vertex three units to the right and two units up. The red figure is the position of the L-shape figure after the slide.

    Figure \(\PageIndex{4}\): L-Shape and Vector


    P

    Figure \(\PageIndex{5}\): Result of the L-Shape Translated by Vector





    P'


    P

    The next type of transformation (rigid motion) that we will discuss is called a rotation. A rotation moves an object about a fixed point R called the rotocenter and through a specific angle. The blue triangle below has been rotated 90° about point R.

    A rotation of an object moves the object around a point called the rotocenter R a certain angle either clockwise or counterclockwise.

    Note: the rotocenter R can be outside the object, inside the object or on the object.

    Figure \(\PageIndex{6}\): A Triangle Rotated 90° around the Rotocenter R outside the Triangle


    90°

    R

    Figure \(\PageIndex{7}\): A Triangle Rotated 180° around the Rotocenter R inside the Triangle


    R

    Properties of a Rotation

    1. A Rotation is completely determined by two pairs of points; P and P’ and

    Q and Q’

    1. Has one fixed point, the rotocenter R
    2. Has identity motion the 360° rotation

    Example \(\PageIndex{3}\): Rotation of an L-Shape

    Given the diagram below, rotate the L-shaped figure 90° clockwise about the rotocenter R. The point Q rotates 90°. Move each vertex 90° clockwise.

    Figure \(\PageIndex{8}\): L-Shape and Rotocenter R

    The L-shaped figure will be rotated 90° clockwise and vertex Q will move to vertex Q'. Each vertex of the object will be rotated 90°.


    Q 90° Q'


    R

    Figure \(\PageIndex{9}\): Result of the 90° Clockwise Rotation



    Q Q'



    R

    Example \(\PageIndex{4}\): 45° Clockwise Rotation of a Rectangle

    Figure \(\PageIndex{10}\): Rectangle and Rotocenter R



    Q 45°

    Q'
    R

    Figure \(\PageIndex{11}\): Result of 45° Clockwise Rotation



    Q

    Q'
    R

    Example \(\PageIndex{5}\): 180° Clockwise Rotation of an L-Shape

    Figure \(\PageIndex{12}\): L-Shape and Rotocenter R

    A





    B R 180°

    Figure \(\PageIndex{13}\): Result of the 180° Clockwise Rotation

    A




    B R
    B'

    A'

    The next type of transformation (rigid motion) is called a reflection. A reflection is a mirror image of an object, or can be thought of as “flipping” an object over.

    Reflection: If each point on a line corresponds to itself, and each other point in the plane corresponds to a unique point in the plane, such that is the perpendicular bisector of, then the correspondence is called the reflection in line .

    In regular language, a reflection is a mirror image across a line . The line is the midpoint of the line between the two points, P in the original figure and P’ in the reflection. P goes to P’.

    Figure \(\PageIndex{14}\): Reflection of an Object about a Line l

    C


    B



    A
    l

    Figure \(\PageIndex{15}\): Result of the Reflection over Line l

    The reflection places each vertex along a line perpendicular to l and equidistant from l.

    C'
    C
    B'
    B




    A' A
    l

    Properties of a Reflection

    1. A reflection is completely determined by a single pair of points; P and P’
    2. Has infinitely many fixed points: the line of reflection l
    3. Has identity motion the reverse reflection

    Example \(\PageIndex{6}\) Reflect an L-Shape across a Line l

    Figure \(\PageIndex{16}\): L-shape and Line l

    B


    C

    A
    l

    Reflect the L-shape across line l. The red L-shape shown below is the result after the reflection. The original position of each vertex is on a line with the reflected position of each vertex. This line that connects the original and reflected positions of the vertex is perpendicular to line l and the original and reflected positions of each vertex are equidistant to line l.

    Figure \(\PageIndex{17}\): Result of Reflection over Line l


    B'

    C'



    l
    A'

    Example \(\PageIndex{7}\): Reflect another L-Shape across Line l

    First identify the vertices of the figure. From each vertex, draw a line segment perpendicular to line l and make sure its midpoint lies on line l. Now draw the new positions of the vertices, making the transformed figure a mirror image of the original figure.

    Figure \(\PageIndex{18}\): L-Shape and Line l


    B
    A
    l
    C
    D




    Figure \(\PageIndex{19}\): Result of Reflection over Line l


    B
    A
    C
    D
    C' B'




    A'

    D'

    The final transformation (rigid motion) that we will study is a glide-reflection, which is simply a combination of two of the other rigid motions.

    A glide-reflection is a combination of a reflection and a translation.

    Example \(\PageIndex{8}\) Glide-Reflection of a Smiley Face by Vector and Line l

    Figure \(\PageIndex{20}\): Smiley Face, Vector , and Line l




    l

    Figure \(\PageIndex{21}\): Smiley Face Glide-Reflection Step One

    First slide the smiley face two units to the right along the vector .






    l

    Figure \(\PageIndex{22}\): Smiley Face Glide-Reflection Step Two

    Then reflect the smiley face across line l. The final result is the green upside-down smiley face.







    l

    Properties of a Glide-Reflection

    1. A reflection is completely determined by a single pair of points; P and P.
    2. Has infinitely fixed points: the line of reflection l.
    3. Has identity motion the reverse glide-reflection.

    Example \(\PageIndex{9}\): Glide-Reflection of a Blue Triangle

    Figure \(\PageIndex{23}\): Blue Triangle, Vector , and Line l

    l



    Figure \(\PageIndex{24}\): Triangle Glide-Reflection Step One

    First, slide the triangle along vector .


    l


    P*


    P

    Figure \(\PageIndex{25}\): Triangle Glide-Reflection Step Two

    Then, reflect the triangle across line l. The final result is the green triangle below line l.

    Q*

    P*
    S*

    P’

    S’ Q’

    Example \(\PageIndex{10}\): Glide-Reflection of an L-Shape

    Figure \(\PageIndex{26}\): L-Shape, Vector , and Line l



    l

    Figure \(\PageIndex{27}\): L-Shape Glide-Reflection Step One

    First slide the L-shape along vector .


    B*



    B
    A*

    A

    Figure \(\PageIndex{28}\): L-Shape Glide-Reflection Step Two

    Then reflect the L-shape across line l. The result is the green open shape below the line l.


    B*
    A’
    B


    B’

    This page titled 10.1: Transformations Using Rigid Motions is shared under a not declared license and was authored, remixed, and/or curated by Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.