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A transformation is a function that moves or changes a figure. The original figure can be called a preimage and the produced figure, an image. Sometimes, transformations are expressed as a mapping. That is, it maps the inputs to the outputs. $T(x,y) \rightarrow (x',y')$

Here, $T$ is the transformation, $(x,y)$ are the coordinates of the points of the preimage, and $(x',y')$ are the coordinates of the points of the image.
A rigid motion is a transformation that preserves the length and angle measures of the preimage. In other words, it maintains the exact size and shape of a figure. Translations, reflections and rotations are rigid transformations.

A translation of a figure is the same as moving it from one position to another in a coordinate plane.

In a translation all points in the preimage are moved the same distance and in the same direction. A triangle that is translated according to the rule $(x,y)\rightarrow (x+4,y+5)$ can look like this.
Each point in the figure is translated $4$ units to the right and $5$ units up.

A composite translation can be seen as the combination of two or more consecutive translations. For example, consider the translations of $PQ \rightarrow P'Q' \rightarrow P''Q''.$

Combining these translations creates the composite translation that directly maps $PQ \rightarrow P''Q''.$

Translate the figure below using the rule $(x,y)\rightarrow (x+4,y-3).$

To begin, notice that the mapping translates the figure in both the horizontal and vertical directions. We can perform each translation separately. First, the horizontal translation $x+4$ moves the figure $4$ units to the right.

The vertical translation, $y-3,$ shifts the figure down $3$ units.

We can now construct the image by connecting the new points with line segments.

Determine the translation of the quadrilateral: $ABCD\rightarrow A'B'C'D'.$

To begin, it might simplify the solution to see the translation as two separate translations, one in the vertical direction and one in the horizontal. We'll identify the mapping of $D\rightarrow D'.$

It can be seen that $D$ is translated $\text{-} 3$ in the vertical direction and $\text{-} 6$ in the horizontal direction. Thus, the mapping can be expressed as$(x,y)\rightarrow (x-6,y-3).$ We can verify this rule by checking the movement of the other points.

When the same rule is applied to the other points, it can be seen that $ABCD\rightarrow A'B'C'D'.$ Thus, the rule for this transformation is $(x,y)\rightarrow(x-6,y-3).$

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