Lesson 14

Defining Rotations

  • Let’s rotate shapes precisely.

14.1: Math Talk: Comparing Angles

For each figure, which pair of angles appears congruent? How could you check?

Figure 1

3 angles. Angle A B C opens to the right, angles D E F and G H L open up.

Figure 2

3 angles. Angles M Z Y and P B K open up, angle R S L opens to the right.

Figure 3

Identical circles. Circle V with central angle GVD opens to the right, circle J with central angle LJX  opens to the left and circle N with central angle CNE opens up.

Figure 4

A figure of 3 circles. H. B. E.

14.2: Info Gap: What’s the Point: Rotations

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.
Several points: B, C, D, F, G, H, I, J, L, N, U, V, W.

14.3: Turning into Triangles

  1. Draw a segment. Label the endpoints \(A\) and \(B\).
    1. Rotate segment \(AB\) clockwise around center \(B\) by 90 degrees. Label the new endpoint \(A’\).
    2. Use the Polygon tool to draw triangle \(ABA'\).
    3. What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.
  2. Draw a segment. Label the endpoints \(C\) and \(D\).
    1. Rotate segment \(CD\) counterclockwise around center \(D\) by 30 degrees. Label the new endpoint \(C’\).
    2. Rotate segment \(C’D\) counterclockwise around center \(D\) by 30 degrees. Label the new endpoint \(C’’\).
    3. Use the Polygon tool to draw triangle \(CDC’’\).
    4. What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.


You constructed an equilateral triangle by rotating a given segment around one of its endpoints by a specific angle measure. An equilateral triangle is an example of a regular polygon: a polygon with all sides congruent and all interior angles congruent. Try to construct some other regular polygons with this method.

Summary

A rotation is a transformation with a center, an angle, and a direction (clockwise or counterclockwise).

Here is how a rotation with a center point \(C\), an angle that measures \(t\) degrees, and a counterclockwise direction transforms a point \(P\):

  • The rotation sends point \(P\) to a point \(P'\) on the circle of radius \(CP\).
  • The angle \(PCP'\) measures \(t\) degrees and \(P'\) is counterclockwise around the circle from \(P\).
Two images of angle t degrees. First shows center C, and t degrees angle, congruent line segments PC and P’C and a counterclockwise rotation taking P to P’. Second shows angle t degrees.

If the direction were clockwise instead, then \(P'\) would be clockwise around the circle of radius \(CP\). If \(P\) and \(C\) are in the same place, then the rotation sends \(P\) to \(P'\) on the circle of radius zero, and so points \(P\), \(C\), and \(P'\) are all in the same place.

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.

  • 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.

  • 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 \(t\).