Lesson 13

Incorporating Rotations

Let's draw some transformations.

13.1: Left to Right

The semaphore alphabet is a way to use flags to signal messages. Here's how to signal the letters Z and J. For each, precisely describe a rotation that would take the left hand flag to the right hand flag.

Z

Stick-figure person with L flag in left hand and R flag in right hand. Both arms are on same side of body. Left arm extends horizontally. Right arm extends down and at an angle.

J

Stick-figure person with L flag in left hand and R flag in right hand. Right arm extends straight up. Left arm extends out horizontally from the body.

 

13.2: Turning on a Grid

  1. Rotate \(ABCD\) 90 degrees clockwise around \(Q\).
  2. Rotate \(ABCD\) 180 degrees around \(R\).
  3. Rotate \(HJKLMN\) 120 degrees clockwise around \(O\).
  4. Rotate \(HJKLMN\) 60 degrees counterclockwise around \(P\).

13.3: Translate, Rotate, Reflect

Mai suspects triangle \(ABC\) is congruent to triangle \(DEF\). She thinks these steps will work to show there is a rigid transformation from \(ABC\) to \(DEF\).

  • Translate by directed line segment \(v\).
  • Rotate the image ____ degrees counterclockwise around point \(D\).
  • Reflect that image over line \(DE\).

Draw each image and determine the angle of rotation needed for these steps to take \(ABC\) to \(DEF\).



Mai’s first 2 steps could be combined into a single rotation.

  1. Find the center and angle of this rotation.
  2. Describe a general procedure for finding a center of rotation.

Summary

The 3 rigid motions are reflect, translate, and rotate. Each of these rigid motions can be applied to any figure to create an image that is congruent. To do a rotation, we need to know 3 things: the center, the direction, and the angle. 

Rotate \(ABCD\) 90 degrees clockwise around point \(P\).

Polygons A B C D and A prime B prime C prime D prime on square grid. Point P located 1 unit up and 1 unit to the right of A. 90 degree angle A P A prime is drawn in.

Rotate \(EFG\) 120 degrees counterclockwise around point \(C\).

Triangles E F G and E prime F prime G prime on isometric grid. Point C located 2 units down and to the right of F prime and 2 units down and to the left of F. Angle F prime C F marked 120 degrees.

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 \(t\).