Lesson 4: Making the Moves

Let’s draw and describe translations, rotations, and reflections.

4.1: Reflection Quick Image

Here is an incomplete image. Your teacher will flash the completed image twice. Your job is to complete the image on your copy.

GeoGebra Applet fSaq5Xc5

4.2: Make That Move

Your partner will describe the image of this triangle after a certain transformation. Sketch it here.

GeoGebra Applet n9AM2ZZp

4.3: A to B to C

Here are some figures on an isometric grid. Explore the transformation tools in the tool bar. (Directions are below the applet if you need them.)

  1. Name a transformation that takes Figure A to Figure B. Name a transformation that takes Figure B to Figure C.

  2. What is one sequence of transformations that takes Figure A to Figure C? Explain how you know.

GeoGebra Applet jqvTEgsj

 

Translate

  1. Select the Vector tool.
  2. Click on the original point and then the new point; you should see a vector. 
  3. Select the Translate by Vector tool. 
  4. Click on the figure to translate, and then click on the vector.

Rotate

  1. Select the Rotate around Point tool. 
  2. Click on the figure to rotate, and then click on the center point.
  3. A dialog box will open; type the angle by which to rotate and select the direction of rotation.

Reflect

  1. Select the Reflect about Line tool. 
  2. Click on the figure to reflect, and then click on the line of reflection.

Summary

A move, or combination of moves, is called a transformation. When we do one or more moves in a row, we often call that a sequence of transformations. To distinguish the original figure from its image, points in the image are sometimes labeled with the same letters as the original figure, but with the symbol $’$ attached, as in $A’$ (pronounced “A prime”).

  • A translation can be described by two points. If a translation moves point $A$ to point $A’$, it moves the entire figure the same distance and direction as the distance and direction from $A$ to $A’$. The distance and direction of a translation can be shown by an arrow.

    For example, here is a translation of quadrilateral $ABCD$ that moves $A$ to $A’$.

    A quadrilateral A, B, C, D, and its translation to A prime, B prime, C prime, D prime.
  • A rotation can be described by an angle and a center. The direction of the angle can be clockwise or counterclockwise.

    For example, hexagon $ABCDEF$ is rotated $90^\circ$ counterclockwise using center $P$.

    A hexagon A, B, C, D, E, F, and its rotation 90 degrees bout a center, P, to hexagon A prime, B prime, C prime, D prime, E prime, F prime.
  • A reflection can be described by a line of reflection (the “mirror”). Each point is reflected directly across the line so that it is just as far from the mirror line, but is on the opposite side.

    For example, pentagon $ABCDE$ is reflected across line $m$.

    A pentagon A, B, C, D, E, and its reflection in a line m, to pentagon A prime, B prime, C prime, D prime, E prime.

Practice Problems ▶

Glossary

transformation

transformation

A transformation is a translation, rotation, reflection, or dilation, or combination of these. There is also a more general concept of a transformation of the plane that is not discussed in grade 8.

sequence of transformations

sequence of transformations

A sequence of transformations is a set of translations, rotations, reflections, and dilations performed in a particular order on a geometric figure, resulting in a final figure. 

The diagram shows a sequence of transformations consisting of a translation (from A to B) followed by a rotation (from B to C) followed by a reflection (from C to D). The last triangle is the final figure resulting from the sequence.