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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.

### Watch This

First watch this video to learn about translations.

CK-12 Foundation Chapter10TranslationsA

Then watch this video to see some examples.

CK-12 Foundation Chapter10TranslationsB

### Guidance

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 ABCD\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 (ABCD)\begin{align*}(A^\prime B^\prime C^\prime D^\prime)\end{align*}. The original square (ABCD)\begin{align*}(ABCD)\end{align*} is called a preimage. The final square is called the image.

#### Example A

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

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

#### Example B

Describe the translation of the light blue triangle in the diagram to the right.

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

#### Example C

Describe the translation in the diagram below.

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

#### Concept Problem Revisited

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 STUV\begin{align*}STUV\end{align*} has moved 13 units to the left and 2 units up.

### Vocabulary

Image
In a transformation, the final figure is called the image.
Preimage
In a transformation, the original figure is called the preimage.
Transformation
A transformation is an operation that is performed on a shape that moves or changes it in some way. There are four types of transformations: translations, reflections, dilations and rotations.
Translation
A translation is an example of a transformation that moves each point of a shape the same distance and in the same direction. Translations are also known as slides.

### Guided Practice

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

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

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

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

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

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

### Practice

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 A(2,3),B(6,3)\begin{align*} A (2, 3), B (6, 3)\end{align*} and C(6,1)\begin{align*}C (6,1)\end{align*} to form ABC\begin{align*}\triangle ABC\end{align*}.

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

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

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

### Vocabulary Language: English

Image

Image

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

Preimage

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

Transformation

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

Translation

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

Rigid Transformation

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