Exploring Translations

A translation is a type of transformation. Other transformations include reflections, rotations, and dilations.

The result of a transformation is called the image. The original figure is called the pre-image.

Use the link below to explore translations:

1. Click on the TRANSLATION button on the left.
2. Click and drag the arrow head and observe what happens.
3. Click and drag the pre-image (the red triangle) and observe what happens.
4. Click and drag a vertex of the pre-image and observe what happens.

After you have explored several translations, answer the following questions in your math journal. Use the words pre-image and image in your responses.

1. When you dragged the arrow head, how did the image compare to the pre-image? What stayed the same? What changed?
2. When you dragged the pre-image, how did the image compare to the pre-image? What stayed the same? What changed?
3. How did dragging the pre-image compare to dragging the arrow head? How were these translations alike? How were they different?
4. When you dragged a vertex of the pre-image, how did the image compare to the pre-image? What stayed the same? What changed?
5. Use the word bank to complete the sentences below:
   1. A translation is a transformation the changes the ________________ of a figure. A translation does not change the figure’s _________ or ________________.
   2. The result of a translation is called the ________. The ____________ figure is called the pre-image.
   3. For any translation, the image and pre-image are ______________.

WORD BANK:

Image, size, congruent, orientation, original, position

Translations on a Coordinate Plane

Record the coordinates of the vertices of a translated polygon as follows:

• Use a capital letter for each vertex of the pre-image. For example, a triangle could have vertices at points A, B, and C.
• Use corresponding capital letters to identify corresponding vertices of the image, adding a prime symbol (‘). For example, the image of the triangle with vertices A, B, and C would have vertices A’, B’, and C’.
• The vertices of the image are read as follows: “A prime,” “B prime,” and “C prime.”

Watch this video to observe the translation of a right triangle.

Respond to or complete the following in your math journal:

1. Summarize the two methods used in the video for translating the triangle.
2. How did the x coordinates of the image compare to the corresponding x coordinates of the pre-image?
3. How did the y-coordinates of the image compare to the corresponding y coordinates of the pre-image?
4. The translation in the video of 9 units to the right and 4 units down could be represented using the rule (x + 9, y – 4). Describe the translation represented by the rule (x – 2, y + 3).

Generating Translations

Use the link below to explore translations of irregular polygons:

1. Click on the ACTIVITIES button at the top of the screen.
2. Follow the directions for “Playing with Translations” that appear on the right side of the screen.
   • Check the box near the bottom left to turn the axes on.
   • Drag the pre-image and/or the arrow head to illustrate translations represented by each of the following rules:
     1. (x + 5, y – 5)
     2. (x – 2, y + 7)
     3. (x – 4, y – 6)
3. Click on either the left or right arrow near the top right of the screen. Follow the directions for “Hitting a Target” that appear on the right side of the screen.
   • Check the box near the bottom left to turn the axes on
   • Determine the approximate rule that could be used to describe this translation: (_________, _________)