8.5: Geometric Translation

Understand translations as movement of every point in a figure the same distance in the same direction. Graph images given preimage and translation.

Translations

A transformation is an operation that moves, flips, or otherwise changes a figure to create a new figure. A rigid transformation (also known as an isometry or congruence transformation) is a transformation that does not change the size or shape of a figure.

The rigid transformations are translations, reflections, and rotations. The new figure created by a transformation is called the image. The original figure is called the preimage. If the preimage is $A$, then the image would be $A′$, said “a prime.” If there is an image of $A′$, that would be labeled $A′′$, said “a double prime.”

A translation is a transformation that moves every point in a figure the same distance in the same direction. For example, this transformation moves the parallelogram to the right 5 units and up 3 units. It is written $(x,y)\rightarrow (x+5, y+3)$.

Figure $\PageIndex{1}$

What if you were given the coordinates of a quadrilateral and you were asked to move that quadrilateral 3 units to the left and 2 units down? What would its new coordinates be?

Example $\PageIndex{1}$

Triangle $\Delta ABC$ has coordinates $A(3,−1)$, $B(7,−5)$ and $C(−2,−2)$. Translate $\Delta ABC$ to the left 4 units and up 5 units. Determine the coordinates of $\Delta A′B′C′$.

Figure $\PageIndex{2}$

Solution

Graph $\Delta ABC$. To translate $\Delta ABC$, subtract 4 from each $x$ value and add 5 to each $y$ value of its coordinates.

$\begin{aligned} &A(3,−1)\rightarrow (3−4,−1+5)=A′(−1,4) \\ &B(7,−5)\rightarrow (7−4,−5+5)=B′(3,0) \\ &C(−2,−2)\rightarrow (−2−4,−2+5)=C′(−6,3) \end{aligned}$

The rule would be $(x,y)\rightarrow (x−4, y+5)$.

Example $\PageIndex{2}$

Using the translation $(x,y)\rightarrow (x+2, y−5)$, what is the image of $A(−6, 3)$?

Solution

$A′(−4,−2)$

Example $\PageIndex{3}$

Graph square $S(1,2)$, $Q(4,1)$, $R(5,4)$ and $E(2,5)$. Find the image after the translation $(x,y)\rightarrow (x−2,y+3)$. Then, graph and label the image.

Solution

We are going to move the square to the left 2 and up 3.

Figure $\PageIndex{3}$

$\begin{aligned}(x,y)&\rightarrow (x−2,y+3) \\ S(1,2)&\rightarrow S′(−1,5) \\ Q(4,1)&\rightarrow Q′(2,4) \\ R(5,4)&\rightarrow R′(3,7) \\ E(2,5)&\rightarrow E′(0,8)\end{aligned}$

Example $\PageIndex{4}$

Find the translation rule for $\Delta TRI$ to $\Delta T′R′I′$.

Solution

Look at the movement from $T$ to $T′$. The translation rule is $(x,y)\rightarrow (x+6, y−4)$.

Figure $\PageIndex{4}$

Figure $\PageIndex{5}$

Review

Use the translation $(x,y)\rightarrow (x+5, y−9)$ for questions 1-7.

1. What is the image of $A(−1,3)$?
2. What is the image of $B(2,5)$?
3. What is the image of $C(4,−2)$?
4. What is the image of $A′$?
5. What is the preimage of $D′(12,7)$?
6. What is the image of $A′′$?
7. Plot $A$, $A′$, $A′′$, and $A′′′$ from the questions above. What do you notice?

The vertices of $\Delta ABC$ are $A(−6,−7)$, $B(−3,−10)$ and $C(−5,2)$. Find the vertices of $\Delta A′B′C′$, given the translation rules below.

8. $(x,y)\rightarrow (x−2, y−7)$
9. $(x,y)\rightarrow (x+11, y+4)$
10. $(x,y)\rightarrow (x, y−3)$
11. $(x,y)\rightarrow (x−5, y+8)$
12. $(x,y)\rightarrow (x+1, y)$
13. $(x,y)\rightarrow (x+3, y+10)$

In questions 14-17, $\Delta A′B′C′$ is the image of $\Delta ABC$. Write the translation rule.

14. 

    Figure $\PageIndex{6}$

15. 

    Figure $\PageIndex{7}$

16. 

    Figure $\PageIndex{8}$

17. 

    Figure $\PageIndex{9}$

Use the triangles from #17 to answer questions 18-20.

18. Find the lengths of all the sides of $\Delta ABC$.
19. Find the lengths of all the sides of $\Delta A′B′C′$.
20. What can you say about $\Delta ABC$ and $\Delta A′B′C′$? Can you say this for any translation?
21. If $\Delta A′B′C′$ was the preimage and $\Delta ABC$ was the image, write the translation rule for #14.
22. If $\Delta A′B′C′$ was the preimage and $\Delta ABC$ was the image, write the translation rule for #15.
23. Find the translation rule that would move $A$ to $A′(0,0)$, for #16.
24. The coordinates of $\Delta DEF$ are $D(4,−2)$, $E(7,−4)$ and $F(5,3)$. Translate $\Delta DEF$ to the right 5 units and up 11 units. Write the translation rule.
25. The coordinates of quadrilateral $QUAD$ are $Q(−6,1)$, $U(−3,7)$, $A(4,−2)$ and $D(1,−8)$. Translate $QUAD$ to the left 3 units and down 7 units. Write the translation rule.

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.3.

Additional Resources

Interactive Element

Video: Transformation: Translation Principles - Basic

Activities: Translations Discussion Questions

Study Aids: Types of Transformations Study Guide

Practice: Geometric Translation