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Translations and Vectors

Graphical introduction to image translations

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Translations

Karen looked at the image below and stated that the image was translated thirteen units backwards. Is she correct? Explain.

Translations

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A translation is a type of transformation that moves each point in a figure the same distance in the same direction. Translations are often referred to as slides. If you look at the picture below, you can see that the square \begin{align*}ABCD\end{align*} is moved 10 units to the right. All points of the square have been moved 10 units to the right to make the translated image \begin{align*}(A^\prime B^\prime C^\prime D^\prime)\end{align*}. The original square \begin{align*}(ABCD)\end{align*} is called a preimage. The final square is called the image.

     

 

Let's describe the following translations:

  1. The preimage is the brown pentagon and the image is the purple pentagon. 

The pentagon is translated down 8 and over 11 to the right.

  1. The preimage is the light blue triangle and the image is the green triangle.

The blue triangle moves up 3 units and over 2 units to the left to make the green triangle image.

  1. The preimage is the purple shape and the image is the yellow shape.

The original shape is translated down 2 and over 7 to the left.

Examples

Example 1

Earlier, you were told that Karen looked at the image below and stated that the image was translated thirteen units backwards. Is she correct? Explain.

Karen is somewhat correct in that the translation is moving to the left (backwards). The proper way to describe the translation is to say that the image \begin{align*}STUV\end{align*} has moved 13 units to the left and 2 units up.

Example 2

Describe the translation of the pink triangle in the diagram below.

The pink triangle is translated down 4 and over 2 to the left.

Example 3

Describe the translation of the purple polygon in the diagram below.

The purple polygon is translated up 2 and over 12 to the right.

Example 4

Describe the translation of the blue hexagon in the diagram below.

 The blue hexagon is translated down 2 and over 10 to the left.

Review

Describe the translation of the purple original figures in the diagrams:

Use the diagram below to describe the following translations:

  1. A onto B
  2. A onto C
  3. A onto D
  4. A onto E
  5. A onto F

On a piece of graph paper, plot the points \begin{align*} A (2, 3), B (6, 3)\end{align*} and \begin{align*}C (6,1)\end{align*} to form \begin{align*}\triangle ABC\end{align*}.

  1. Translate the triangle 3 units to the right and 2 units down.                                            Label this \begin{align*}\triangle A^\prime B^\prime C^\prime\end{align*}.
  2. Translate \begin{align*}\triangle A^\prime B^\prime C^\prime\end{align*} 3 units to the left and 4 units down.                                                 Label this \begin{align*}\triangle A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\end{align*}.
  3. Describe the translation necessary to bring \begin{align*}\triangle A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\end{align*} to \begin{align*}\triangle ABC\end{align*}.

On a piece of graph paper, plot the points \begin{align*} D (1, 5), E (2, 3)\end{align*} and \begin{align*}F (1,0)\end{align*} to form \begin{align*}\triangle ABC\end{align*}.

  1. Translate the triangle 2 units to the left and 4 units down.                                               Label this \begin{align*}\triangle D^\prime E^\prime F^\prime\end{align*}.
  2. Translate \begin{align*}\triangle D^\prime E^\prime F^\prime\end{align*} 5 units to the right and 2 units up.                                                   Label this \begin{align*}\triangle D^{\prime \prime} E^{\prime \prime} F^{\prime \prime}\end{align*}.
  3. Describe the translation necessary to bring \begin{align*}\triangle D^{\prime \prime} E^{\prime \prime} F^{\prime \prime}\end{align*} to \begin{align*}\triangle DEF\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.1. 

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Vocabulary

Image

The image is the final appearance of a figure after a transformation operation.

Preimage

The pre-image is the original appearance of a figure in a transformation operation.

Transformation

A transformation moves a figure in some way on the coordinate plane.

Translation

A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

Image Attributions

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