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 \begin{align*}\vec{v}\end{align*} to create a new parallelogram (the image).
Keep in mind that the location of vector @$\begin{align*}\vec{v}\end{align*}@$ does not matter. Vectors have a direction and a magnitude (a length), and simply tell you how to move points. Vector @$\begin{align*}\vec{v}\end{align*}@$ 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 @$\begin{align*}\vec{v}\end{align*}@$.
With the grid in the background, you can see that the slope of vector @$\begin{align*}\vec{v}\end{align*}@$ and each line is @$\begin{align*}\frac{1}{3}\end{align*}@$. 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 @$\begin{align*}\sqrt{10}\end{align*}@$ (found using the Pythagorean Theorem) along lines parallel to vector @$\begin{align*}\vec{v}\end{align*}@$.
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 @$\begin{align*}\vec{v}\end{align*}@$ have been drawn through each of the original points. Vector @$\begin{align*}\vec{v}\end{align*}@$ 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 |
@$\begin{align*}A\end{align*}@$ to @$\begin{align*}A^\prime\end{align*}@$ |
4 to the right and 1 up |
@$\begin{align*}B\end{align*}@$ to @$\begin{align*}B^\prime\end{align*}@$ |
4 to the right and 1 up |
@$\begin{align*}C\end{align*}@$ to @$\begin{align*}C^\prime\end{align*}@$ |
4 to the right and 2 up |
@$\begin{align*}D\end{align*}@$ to @$\begin{align*}D^\prime\end{align*}@$ |
4 to the right and 1 up |
Because @$\begin{align*}C\end{align*}@$ to @$\begin{align*}C^\prime\end{align*}@$ 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 @$\begin{align*}\vec{u}\end{align*}@$ on the quadrilateral below.
Solution: With the grid in the background, you can see that vector @$\begin{align*}\vec{u}\end{align*}@$ 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 @$\begin{align*}\vec{t}\end{align*}@$ on the triangle below.
3. Is the following transformation a translation?
Answers:
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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Perform the translation defined by vector @$\begin{align*}\vec{t}\end{align*}@$ on the polygons below.
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Are the following transformations translations? Explain.
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