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Here is some recommended reading before getting started with this lesson.
From the exploration, the following conclusions can be drawn.
To perform a translation, a vector is required. This implies that the direction of the translation plays an important role. To illustrate this statement, consider the following diagram.
It can be seen that the lengths of $AA_{′},$ $BB_{′},$ and $CC_{′}$ are equal to the magnitude of $v.$ Also, $AA_{′},$ $BB_{′},$ and $CC_{′}$ are parallel to $v.$ Therefore, the same two conclusions written above apply to this diagram.Consequently, and referring to what can be inferred from the exploration applet, a third conclusion can be drawn.
The vectors $AA_{′},$ $BB_{′},$ and $CC_{′}$ have the same direction as $v.$
Be aware that the three conclusions written before do not depend on the preimage. They hold true even when the preimage is a single point, a segment, a polygon, or any other figure. Then, these properties can be used to define a translation properly.
A translation is a transformation that moves every point of a figure the same distance in the same direction. More precisely, a translation along a vector $v$ maps every point $A$ in the plane onto its image $A_{′}$ such that the following statements hold true.
As the previous exploration shows, translations preserve side lengths and angle measures. That confirms translations are rigid motions. Additionally, take note that translations map segments onto parallel segments. Consider the polygon $P$ shown in the middle of the diagram below. The other polygons are images of $P$ after different transformations.
What polygons are the image of $P$ after a translation?
Only polygons $P_{4},$ $P_{6},$ and $P_{14}$ are the image of $P$ after a translation.
Remember that translations map segments onto parallel segments. Since $P$ looks like an arrow pointing up, the images of $P$ after a translation will also look like an arrow pointing up.
Since translations map segments onto parallel segments, the sides of $P$ are parallel to the sides of its image after any translation. Notice that $P$ looks like an arrow pointing up.
Therefore, the image of $P$ after any translation will also look like an arrow pointing up. In the given diagram, it can be seen that only $P_{4},$ $P_{6},$ and $P_{14}$ satisfy this condition. Consequently, these polygons are images of $P$ after a translation.In the previous example, only the shape of polygon $P$ was used to determine which polygons were the image of $P$ after a translation. When the vertices are labeled, keep an eye on them. Consider the following three squares.
Which of the squares $S_{1}$ or $S_{2},$ if either, is the image of $PQRS$ after a translation?Therefore, $S_{2}$ is not the image of $PQRS$ after a translation. Conversely, $S_{1}$ satisfies the two conditions previously written. Even more, since every vector that connects a vertex to its image have the same direction, it can be concluded that $S_{1}$ is a translation of $PQRS.$
Translations can be performed by hand with the help of a straightedge and a compass.
To translate $△ABC$ along $v$ follow the four steps below.
Draw parallel lines to $ℓ$ through each vertex.
Draw the triangle formed by the image points $A_{′},$ $B_{′},$ and $C_{′}.$
By definition of translation, vectors $AA_{′}$ and $v$ are parallel and have the same magnitude and direction. This makes $PAA_{′}Q$ a parallelogram. With this in mind, the translation of $△ABC$ can be performed as follows.
In the following applet, one of the following tasks may be required.
To translate $△ABC,$ place points $A_{′},$ $B_{′},$ and $C_{′}$ where they should be after the translation is applied.
When learning about rotations, it was said that the composition of two rotations could be a translation. Now, the composition of two translations will be examined.
Consider the following pair of quadrilaterals $P_{1}$ and $P_{2}.$ Also, consider a pair of different translations. One translation along vector $u=⟨5,1⟩$ and the other along $v=⟨-2,-4⟩.$
Translating Along | Is Equivalent To |
---|---|
$u=⟨5,1⟩$ | Translating $5$ units to the right and $1$ unit up. |
$v=⟨-2,-4⟩$ | Translating $2$ units to the left and $4$ unit down. |
All the vectors drawn seem to be parallel and with the same magnitude. Even more, they all have the same direction. This could be checked by finding the component form of each vector.
Vertex | Image | Vector | Component Form |
---|---|---|---|
$(-4,0)$ | $(-1,-3)$ | $⟨-1−(-4),-3−0⟩$ | $⟨3,-3⟩$ |
$(-1,1)$ | $(2,-2)$ | $⟨2−(-1),-2−1⟩$ | $⟨3,-3⟩$ |
$(-1,2)$ | $(2,-1)$ | $⟨2−(-1),-1−2⟩$ | $⟨3,-3⟩$ |
$(-3,3)$ | $(0,0)$ | $⟨0−(-3),0−3⟩$ | $⟨3,-3⟩$ |
As the table shows, all the vectors connecting a preimage with its image have the same component form. This confirms that the vectors are parallel and have the same magnitude and direction. Consequently, a translation along $⟨3,-3⟩$ maps $P_{1}$ onto $P_{2}.$
The two conclusions obtained in the previous example are not a coincidence. In fact, these are general results when performing a composition of translations.
Be aware that a composition of transformations might involve translations and rotations. This combination can produce interesting images.
In interior design, it is pretty common to see designs consisting of a single preimage and its images under different transformations such as translations and rotations. Below, two different kitchen tile designs are made using just four right triangles.