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# Translations and Vectors

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Translations

Translations are often informally called “slides”. Why is this?

#### Guidance

A translation is one example of a rigid transformation . A translation moves each point in a shape a specified distance in a specified direction as defined by a vector. Below, the parallelogram has been translated along vector $\vec{v}$  to create a new parallelogram (the image).

Keep in mind that the location of vector $\vec{v}$   does not matter . Vectors have a direction and a magnitude (a length), and simply tell you how to move points. Vector  $\vec{v}$ essentially tells you that all points move three units to the right and one unit up

The lines that connect corresponding points will all be parallel to vector $\vec{v}$ .

With the grid in the background, you can see that the slope of vector  $\vec{v}$ and each line is $\frac{1}{3}$ . This should make sense because each point in the original parallelogram was moved 3 units to the right and 1 unit up to create its corresponding point in the image.

One way to think about translations is that they move points a specified distance along lines parallel to a given line.  In this case, all points were moved a distance of $\sqrt{10}$  (found using the Pythagorean Theorem) along lines parallel to vector $\vec{v}$ .

If you are performing a translation of a shape not on a grid, the vector becomes crucial. You can no longer say “move one unit up and three units to the right” because without a grid there are no units. Below is a translation of another quadrilateral without a grid in the background.

Notice that lines parallel to vector  $\vec{v}$ have been drawn through each of the original points. Vector  $\vec{v}$ has been copied onto each of those lines at the points that define the original quadrilateral. The ends of the vectors define the points on the image.

Example A

Is the following transformation a translation?

Solution: One way to check if a transformation is a translation is to look at how each point moves to create its image. If all points move in the same way, then it is a translation.

 Point to Image Point Description of Motion $A$  to $A^\prime$ 4 to the right and 1 up $B$  to $B^\prime$ 4 to the right and 1 up $C$  to $C^\prime$ 4 to the right and 2 up $D$  to $D^\prime$ 4 to the right and 1 up

Because  $C$ to  $C^\prime$ is different, this is not a translation.

Example B

Describe the vector that defined the translation below.

Solution: The vector moved each point three units to the right and three units up.

Example C

Perform the translation defined by vector  $\vec{u}$ on the quadrilateral below.

Solution: With the grid in the background, you can see that vector  $\vec{u}$ tells you to move each point 2 units up and 1 unit to the left. Here is the translation:

Concept Problem Revisited

A translation is informally called a slide, because it essentially slides a shape to a new position. The orientation of the points does not change.

#### Vocabulary

A translation is a rigid transformation that moves each point in a shape a specified distance in a specified direction as defined by a vector.

#### Guided Practice

1. Describe the vector that defined the translation below.

2. Perform the translation defined by vector  $\vec{t}$ on the triangle below.

3. Is the following transformation a translation?

1. The vector moved each point 4 units to the right and 2 units down.

2.

3. Yes, each point moves one unit to the right and two units up.

#### Practice

1. Is a translation a rigid transformation? Explain.

2. What role does a vector play in a translation?

3. How are parallel lines relevant to translations?

4. How can you tell if a transformation is a translation?

Describe the vector that defined each of the following translations.

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6.

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8.

Perform the translation defined by vector  $\vec{t}$ on the polygons below.

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12.

Are the following transformations translations? Explain.

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