Lesson 11

Defining Reflections

  • Let’s reflect some figures.

11.1: Which One Doesn’t Belong: Crossing the Line

Which one doesn’t belong?

Figure 1

Vertical line l intersects line A B.  Line AB is slanted downward and to the right. The segment between point A and line l is labeled 4, the segment between line l and Point B is labeled 6.

Figure 2

Vertical lines l intersects horizontal line A B. The segment between point A and line l is labeled 4, the segment between line l and point B is labeled 6. 

Figure 3

Vertical line L intersects horizontal line C C prime. On line C C prime, two tick marks labeled 5, on either side of line L.

Figure 4

Line l is the perpendicular bisector of AB, each half of AB is 7.

 

11.2: Info Gap: What’s the Point: Reflections

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

  1. Silently read the information on your card.
  2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
  3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
  4. Read the problem card, and solve the problem independently.
  5. Share the data card, and discuss your reasoning.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to answer the question.
  2. Ask your partner for the specific information that you need.
  3. Explain to your partner how you are using the information to solve the problem.
  4. When you have enough information, share the problem card with your partner, and solve the problem independently.
  5. Read the data card, and discuss your reasoning.
14 points on a circle.

11.3: Triangle in the Mirror

Kiran started reflecting triangle \(CDE\) across line \(m\). So far, he knows the image of \(D\) is \(D’\) and the image of \(E\) is \(E’\).

  1. Annotate the diagram to show how he reflected point \(D\).
  2. Use straightedge and compass moves to determine the location of \(C’. \) Then lightly shade in triangle \(C’D’E’\).
  3. Write a set of instructions for how to reflect any point \(P\) across a given line \(\ell\).
  4. Elena found \(C’\) incorrectly:

    Elena's Diagram

    Triangles C D E and C prime D prime E prime.
    Elena is convinced that triangle \(C’D’E’\) “looks fine.” Explain to Elena why her \(C’\) is not a reflection of point \(C\) across line \(m\).


  1. Using your response from question 2 (with the correct location of \(C’\)):
    1. Draw the line \(CC’\).
    2. Reflect triangle \(C’D’E’\) across line \(CC'\).
    3. Label the image \(C’’D’’E’’\).
  2. Find a single rigid motion that takes \(CDE\) to \(C’’D’’E’’\).

Summary

Think about reflecting the point \(A\) across line \(\ell\):

Line L, facing downward and to the right. Point A sits to the left of the line.

The image \(A’\) is somewhere on the other side of \(\ell\) from \(A\). The line \(\ell\) is the boundary between all the points that are closer to \(A\) and all the points that are closer to \(A’\). In other words, \(\ell\) is the set of points that are the same distance from \(A\) as from \(A’\). In a previous lesson, we conjectured that a set of points that are the same distance from \(A\) as from \(A’\) is the perpendicular bisector of the segment \(AA’\). Using a construction technique from a previous lesson, we can construct a line perpendicular to \(\ell\) that goes through \(A\):

Line L, facing downward and to the right. A perpendicular line passes through Line L and Point A.

\(A’\) lies on this new line at the same distance from \(\ell\) as \(A\):

Perpendicular lines and a circle

We define the reflection across line \(\ell\) as a transformation that takes each point \(A\) to a point \(A’\) as follows: \(A’\) lies on the line through \(A\) that is perpendicular to \(\ell\), is on the other side of \(\ell\), and is the same distance from \(\ell\) as \(A\). If \(A\) happens to be on line \(\ell\), then \(A\) and \(A’\) are both at the same location (they are both a distance of zero from line \(\ell\)).

Line l is the perpendicular bisector of AA’.

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.

  • 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.