Lesson 12

Defining Translations

  • Let’s translate some figures.

12.1: Notice and Wonder: Two Triangles and an Arrow

What do you notice? What do you wonder?

Triangle C D E, blue, on the right, and unmarked triangle, yellow, on the left. Directed line segment V, facing upwards and to the right, with an arrow at the right end.

 

12.2: What’s the Point: Translations

  1. After a translation, the image of \(V\) is \(W\). Find at least 3 other points that are taken to a labeled point by that translation.
  2. Write at least 1 conjecture about translations.
  3. In a new translation, the image of \(V\) is \(Z\). Find at least 3 other points that are taken to a labeled point by the new translation.
  4. Are your conjectures still true for the new translation?
26 points, labeled A through Z, scattered throughout an area. Point A is at the top right, and point Z is towards the bottom left.

12.3: Translating Triangles

  1. Translate triangle \(ABC\) by the directed line segment from \(A\) to \(C\).
    1. What is the relationship between line \(BC\) and line \(B’C’\)? Explain your reasoning.
    2. How does the length of segment \(BC\) compare to the length of segment \(B’C’\)? Explain your reasoning.
  2. Translate segment \(DE\) by directed line segment \(w\). Label the new endpoints \(D’\) and \(E’\).
    1. Connect \(D\) to \(D’\) and \(E\) to \(E’\).
    2. What kind of shape did you draw? What properties does it have? Explain your reasoning.


  1. On triangle \(ABC\) in the task, use a straightedge and compass to construct the line which passes through \(A\) and is perpendicular to \(AC\). Label it \(\ell\). Then, construct the perpendicular bisector of \(AC\) and label it \(m\). Draw the reflection of \(ABC\) across the line \(\ell\). Since the label \(A’B’C’\) is used already, label it \(DEF\) instead.
  2. What is the reflection of \(DEF\) across the line \(m\)?
  3. Explain why this is cool.

Summary

A translation slides a figure in a given direction for a given distance with no rotation. The distance and direction is given by a directed line segment. The arrow of the directed line segment specifies the direction of the translation, and the length of the directed line segment specifies how far the figure gets translated.

Triangle C D E, Triangle C prime D prime E prime, and directed line segment v. V slants downward and to the left, begins below point D and ends below point D prime.

More precisely, a translation of a point \(A\) along a directed line segment \(t\) is a transformation that takes \(A\) to \(A’\) so that the directed line segment \(AA’\) is parallel to \(t\), goes in the same direction as \(t\), and is the same length as \(t\).

Directed line segments 

Here is a translation of 3 points. Notice that the directed line segments \(CC’\), \(DD’\), and \(EE’\) are each parallel to \(v\), going in the same direction as \(v\), and the same length as \(v\). 

Two triangles and four direct line segments

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.

  • 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 \(m\).

  • rigid transformation

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

  • 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 \(t\).