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Describing Translations

Concept

Transformation

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.
Concept

Rigid Motion

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.
Concept

Translation

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.
Concept

Composite Translations

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 $PQ \rightarrow P''Q''.$

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Exercise

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

Show Solution
Solution

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.

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Exercise

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

Show Solution
Solution

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 $(x,y)\rightarrow(x-6,y-3).$