8th Grade - Translation

Introduction

• There are four different types of transformations.

1. 1. Dilations
   2. Reflections
   3. Rotations
   4. Translations

• To transform a figure (shape) means changing the size, location, and direction it faces.
• The figure before the transformation occurs is called the pre-image.
• The figure after the transformation is called the image.
• If the pre-image 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 pre-image.
• 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, pre-image and image are congruent.
• The figure can slide in any direction on the coordinate plane. 

   

Solved Examples

Question 1: $△A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ is the translated image of $△ABC$. What is the translation rule that models the given transformation?

Solution: From the graph, it is evident that each point of the pre-image is shifted 2 units to the left and 3 units upwards. Hence, the translation rule is $\left(x, y\right)\to \left(x-2, y+3\right)$

 

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 x-coordinate moves from -3 to 1, so the horizontal movement is $\left(x+4\right)$. The y-coordinate moves from 6 to -7, so the vertical movement is $\left(y-13\right)$. Therefore, we want to translate each coordinate as $\left(x+4, y-13\right)$.

The coordinates of $B^{\text{'}}$ are $x=\left(1+4\right)=5$, and $y=\left(4-13\right)=-9$. Point $B^{\text{'}}$ is at $\left(5, -9\right)$.

The coordinates of $C^{\text{'}}$ are $x=\left(-7+4\right)=-3$, and $y=\left(-5-13\right)=-18$. Point $B^{\text{'}}$ is at $\left(-3, -18\right)$.

The coordinates of $D^{\text{'}}$ are $x=\left(-6+4\right)=-2$, and $y=\left(-2-13\right)=-15$. 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 pre-image (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.