Introduction
 There are four different types of transformations.

 Dilations
 Reflections
 Rotations
 Translations
 To transform a figure (shape) means changing the size, location, and direction it faces.
 The figure before the transformation occurs is called the preimage.
 The figure after the transformation is called the image.
 If the preimage is labeled as A, then the image would then be labeled as A' (pronounced as "A prime").
Congruency
 Two images are congruent if they are of the same shape and size.
 The image after translation is congruent to the preimage.
 Translations are rigid motions. Rigid motions preserve distance (side lengths) and angle measures.
 The symbol for congruency is ≅.
Translation
 A translation moves (slides) or displaces every point of the figure by the same distance and in the same direction.
 Translation in coordinate plane: $\left(x,y\right)\to \left(x+a,y+a\right)$
 In translation, the figure slides but never turns or rotates.
 In translation, preimage and image are congruent.
 The figure can slide in any direction on the coordinate plane.
Solved Examples
Question 1: $\u25b3{A}^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ is the translated image of $\u25b3ABC$. What is the translation rule that models the given transformation?
Solution: From the graph, it is evident that each point of the preimage is shifted 2 units to the left and 3 units upwards. Hence, the translation rule is $\left(x,y\right)\to \left(x2,y+3\right)$
Question 2: The coordinates of a quadrilateral are shown in the table below. If the image has ${A}^{\text{'}}$ at a point $\left(1,7\right)$, what are the coordinates of ${B}^{\text{'}}$,${C}^{\text{'}}$ and ${D}^{\text{'}}$?
Solution: To find the coordinates for these points, first identify the translation that moves the point $A$ to ${A}^{\text{'}}$. The xcoordinate moves from 3 to 1, so the horizontal movement is $\left(x+4\right)$. The ycoordinate moves from 6 to 7, so the vertical movement is $\left(y13\right)$. Therefore, we want to translate each coordinate as $\left(x+4,y13\right)$.
The coordinates of ${B}^{\text{'}}$ are $x=\left(1+4\right)=5$, and $y=\left(413\right)=9$. Point ${B}^{\text{'}}$ is at $\left(5,9\right)$.
The coordinates of ${C}^{\text{'}}$ are $x=\left(7+4\right)=3$, and $y=\left(513\right)=18$. Point ${B}^{\text{'}}$ is at $\left(3,18\right)$.
The coordinates of ${D}^{\text{'}}$ are $x=\left(6+4\right)=2$, and $y=\left(213\right)=15$. Point ${B}^{\text{'}}$ is at $\left(2,15\right)$.
Cheat Sheet
 The algebraic representation of a translation depends on the directions of the displacement.
 When the $x$values are changed, the figure moves horizontally (left or right).
 When the $y$values are changed, the figure moves vertically (up or down).
Blunder Areas
 Differentiate between preimage (A) and image (A').
 Think of the movements in terms of positive and negative.
 Movements to the right and up are positive.
 Movements to the left and down are negative.
 Horizontal (left and right) movements change the $x$values.
 Vertical (up and down) movements change the $y$values.
 Fiona Wong
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