From the exploration, the following conclusions can be drawn.
It can be seen that the lengths of and are equal to the magnitude of Also, and are parallel to Therefore, the same two conclusions written above apply to this diagram. However, is not a translation of This is because, although vectors and have the same direction as the vector does not have the same direction as
Consequently, and referring to what can be inferred from the exploration applet, a third conclusion can be drawn.
The vectors and have the same direction as
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 maps every point in the plane onto its image 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 shown in the middle of the diagram below. The other polygons are images of after different transformations.
What polygons are the image of after a translation?
Only polygons and are the image of after a translation.
Remember that translations map segments onto parallel segments. Since looks like an arrow pointing up, the images of after a translation will also look like an arrow pointing up.
Since translations map segments onto parallel segments, the sides of are parallel to the sides of its image after any translation. Notice that looks like an arrow pointing up.
In the previous example, only the shape of polygon was used to determine which polygons were the image of after a translation. When the vertices are labeled, keep an eye on them. Consider the following three squares.
Therefore, is not the image of after a translation. Conversely, 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 is a translation of
To translate along follow the four steps below.
|Translating Along||Is Equivalent To|
|Translating units to the right and unit up.|
|Translating units to the left and 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.
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 maps onto