Lesson 17

Working with Rigid Transformations

  • Let’s compare transformed figures.

17.1: Math Talk: From Here to There

Segment \(CD\) is the perpendicular bisector of segment \(AB\). Find each transformation mentally.

A transformation that takes \(A\) to \(B\).

A transformation that takes \(B\) to \(A\).

A transformation that takes \(C\) to \(D\).

A transformation that takes \(D\) to \(C\).

Segments AB and CD intersect at point M. CD is the perpendicular bisector of AB.

17.2: Card Sort: How Did This Get There?

  1. Your teacher will give you a set of cards that show transformations of figures.

    1. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.

    2. Then sort the cards into categories in a different way. Be prepared to explain the meaning of your new categories.

  2. For each card with a rigid transformation: write a sequence of rotations, translations, and reflections to get from the original figure to the image. Be precise.



Diego observes that although it was often easier to use a sequence of reflections, rotations, and translations to describe the rigid transformations in the cards, each of them could be done with just a single reflection, rotation, or translation. However, Priya draws her own card, shown, which she claims can not be done as a single reflection, rotation, or translation.

Triangles A B C and A prime B Prime C Prime.
  1. For each rigid transformation from the card sort, write the transformation as a single reflection, rotation, or translation.
  2. Justify why Priya’s transformation cannot be written as a single reflection, rotation, or translation.

17.3: Reflecting on Reflection

Diego says, “I see why a reflection could take \(RSTU\) to \(R'S'T'U'\), but I’m not sure where the line of reflection is. I’ll just guess.”

  1. How could Diego see that a reflection could work without knowing where the line of reflection is?
  2. How could Diego find an exact line of reflection that would work?

Summary

If 2 figures are congruent, we can always find a rigid transformation that takes one onto the other.

Look at congruent figures \(ABC\) and \(DEF\). It looks like ​​​​​\(DEF\) might be a reflection and translation of \(ABC\). But is there a way to describe a sequence of transformations without guessing where the line of reflection might be?

Two congruent triangles labeled ABC and DEF.

Our goal is to take the image of \(E\) onto \(B\). Then we want to take the image of \(D\) onto \(A\) without moving \(E\) and \(B\). Finally, we need to take the image of \(F\) onto \(C\) without moving any of the matching points.

We can start with translation: Translate triangle \(DEF\) by the directed line segment from \(E\) to \(B\).

Three congruent triangles, DEF, D’E’F’ and ABC.

 

Now, a pair of corresponding points coincides. Is there a transformation we could use to take \(D'\) onto \(A\) that leaves \(B\) and \(E'\) in place? Rotations have a fixed point, so rotate triangle \(D'E'F'\) by angle \(D'BA\) using point \(B\) as the center.

Four congruent triangles, DEF, D’E’F’, D”E”F” and ABC.

Now, 2 pairs of corresponding points coincide. Reflecting across line \(AB\) will take \(D''E''F''\) onto \(ABC\), which is what we were trying to do. We know \(D''\) and \(E''\) won’t move, since points on the line of reflection don't move. How do we know \(F''\) will end up on \(C\)? Since the triangles are congruent, \(F''\)and \(C\) are the same distance from the line of reflection.

It is always possible to describe transformations using existing points, angles, and segments. It could take an extra step, but we can be confident transformations work if we don't guess where the line of reflection or center of rotation might be.

Glossary Entries

  • assertion

    A statement that you think is true but have not yet proved.

  • congruent

    One figure is called congruent to another figure if there is a sequence of translations, rotations, and reflections that takes the first figure onto the second.

  • directed line segment

    A line segment with an arrow at one end specifying a direction.

  • image

    If a transformation takes \(A\) to \(A'\), then \(A\) is the original and \(A'\) is the image.

  • line of symmetry

    A line of symmetry for a figure is a line such that reflection across the line takes the figure onto itself.

    The figure shows two lines of symmetry for a regular hexagon, and two lines of symmetry for the letter I.

  • reflection

    A reflection is defined using a line. It takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line. 

    In the figure, \(A'\) is the image of \(A\) under the reflection across the line \(m\).

    Reflect $A$ across line $m$.

  • reflection symmetry

    A figure has reflection symmetry if there is a reflection that takes the figure to itself.

  • rigid transformation

    A rigid transformation is a translation, rotation, or reflection. We sometimes also use the term to refer to a sequence of these.

  • rotation

    A rotation has a center and a directed angle. It takes a point to another point on the circle through the original point with the given center. The 2 radii to the original point and the image make the given angle.

    \(P'\) is the image of \(P\) after a counterclockwise rotation of  \(t^\circ\) using the point \(O\) as the center.

    ​​​​​Quadrilateral \(ABCD\) is rotated 120 degrees counterclockwise using the point \(D\) as the center.

  • rotation symmetry

    A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (We don't count rotations using angles such as \(0^\circ\) and \(360^\circ\) that leave every point on the figure where it is.)

  • symmetry

    A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is).

  • theorem

    A statement that has been proved mathematically. 

  • translation

    A translation is defined using a directed line segment. It takes a point to another point so that the directed line segment from the original point to the image is parallel to the given line segment and has the same length and direction.

    In the figure, \(A'\) is the image of \(A\) under the translation given by the directed line segment \(t\).