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# 12.2: Translations

Created by: CK-12

## Learning Objectives

• Graph a point, line, or figure and translate it $x$ and $y$ units.
• Write a translation rule.

## Review Queue

1. Find the equation of the line that contains (9, -1) and (5, 7).
2. What type of quadrilateral is formed by $A(1, -1), B(3, 0), C(5, -5)$ and $D(-3, 0)$?
3. Find the equation of the line parallel to #1 that passes through (4, -3).

Know What? The distances between San Francisco, $S$, Paso Robles, $P$, and Ukiah, $U$, are given in miles the graph. Find:

a) The translation rule for $P$ to $S$.

b) The translation rule for $S$ to $U$.

c) The translation rule for $P$ to $U$.

d) The translation rule for $U$ to $S$. It is not the same as part b.

## Transformations

Transformation: An operation that moves, flips, or changes a figure to create a new figure.

Rigid Transformation: 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. Another word for a rigid transformation is an isometry or congruence transformations.

In Lesson 7.6, we learned how to label an image. If the preimage is $A$, then the image would be $Aâ€™$, said “a prime.” If there is an image of $Aâ€™$, that would be labeled $Aâ€$, said “a double prime.”

## Translations

Translation: A transformation that moves every point in a figure the same distance in the same direction.

This transformation moves the parallelogram to the right 5 units and up 3 units. It is written $(x,y) \rightarrow (x+5,y+3)$.

Example 1: 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.

$(x,y) &\rightarrow (x-2,y+3)\\S(1,2) &\rightarrow Sâ€™(-1,5)\\Q(4,1) &\rightarrow Qâ€™(2,4)\\R(5,4) &\rightarrow Râ€™(3,7)\\E(2,5) &\rightarrow Eâ€™(0,8)$

Example 2: Find the translation rule for $\triangle TRI$ to $\triangle Tâ€™Râ€™Iâ€™$.

Solution: Look at the movement from $T$ to $Tâ€™$. The translation rule is $(x,y) \rightarrow (x+6,y-4)$.

Example 3: Show $\triangle TRI \cong \triangle Tâ€™Râ€™Iâ€™$ from Example 2.

Solution: Use the distance formula to find all the lengths of the sides of the two triangles.

$& \underline{\triangle TRI} && \underline{\triangle Tâ€™Râ€™Iâ€™}\\& TR=\sqrt{(-3-2)^2+(3-6)^2}=\sqrt{34} && Tâ€™Râ€™=\sqrt{(3-8)^2+(-1-2)^2}=\sqrt{34}\\& RI=\sqrt{(2-(-2))^2+(6-8)^2}=\sqrt{20} && Râ€™Iâ€™=\sqrt{(8-4)^2+(2-4)^2}=\sqrt{20}\\& TI=\sqrt{(-3-(-2))^2+(3-8)^2}=\sqrt{26} && Tâ€™Iâ€™=\sqrt{(3-4)^2+(-1-4)^2}=\sqrt{26}$

This verifies our statement at the beginning of the section that a translation is an isometry or congruence translation.

Example 4: Triangle $\triangle ABC$ has coordinates $A(3, -1), B(7, -5)$ and $C(-2, -2)$. Translate $\triangle ABC$ to the left 4 units and up 5 units. Determine the coordinates of $\triangle Aâ€™Bâ€™Câ€™$.

Solution: Graph $\triangle ABC$. To translate $\triangle ABC$, subtract 4 from each $x$ value and add 5 to each $y$ value.

$& A(3,-1) \rightarrow (3-4,-1+5)=Aâ€™(-1,4)\\& B(7,-5) \rightarrow (7-4,-5+5)=Bâ€™(3,0)\\& C(-2,-2) \rightarrow (-2-4,-2+5)=Câ€™(-6,3)$

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

Know What? Revisited

a) $(x,y) \rightarrow (x-84,y+187)$

b) $(x,y) \rightarrow (x-39,y+108)$

c) $(x,y) \rightarrow (x-123,y+295)$

d) $(x,y) \rightarrow (x+39,y-108)$

## Review Questions

• Questions 1-13 are similar to Example 1.
• Questions 14-17 are similar to Example 2.
• Questions 18-20 are similar to Example 3.
• Questions 21-23 are similar to Example 1.
• Questions 24 and 25 are similar to Example 4.

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

1. What is the image of $A(-6, 3)$?
2. What is the image of $B(4, 8)$?
3. What is the image of $C(5, -3)$?
4. What is the image of $Aâ€™$?
5. What is the preimage of $Dâ€™(12, 7)$?
6. What is the image of $Aâ€$?
7. Plot $A, Aâ€™, Aâ€,$ and $Aâ€â€™$ from the questions above. What do you notice?

The vertices of $\triangle ABC$ are $A(-6, -7), B(-3, -10)$ and $C(-5, 2)$. Find the vertices of $\triangle Aâ€™Bâ€™Câ€™$, given the translation rules below.

1. $(x,y) \rightarrow (x-2,y-7)$
2. $(x,y) \rightarrow (x+11,y+4)$
3. $(x,y) \rightarrow (x,y-3)$
4. $(x,y) \rightarrow (x-5,y+8)$
5. $(x,y) \rightarrow (x+1,y)$
6. $(x,y) \rightarrow (x+3,y+10)$

In questions 14-17, $\triangle Aâ€™Bâ€™Câ€™$ is the image of $\triangle ABC$. Write the translation rule.

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

1. Find the lengths of all the sides of $\triangle ABC$.
2. Find the lengths of all the sides of $\triangle Aâ€™Bâ€™Câ€™$.
3. What can you say about $\triangle ABC$ and $\triangle Aâ€™Bâ€™Câ€™$? Can you say this for any translation?
4. If $\triangle Aâ€™Bâ€™Câ€™$ was the preimage and $\triangle ABC$ was the image, write the translation rule for #14.
5. If $\triangle Aâ€™Bâ€™Câ€™$ was the preimage and $\triangle ABC$ was the image, write the translation rule for #15.
6. Find the translation rule that would move $A$ to $Aâ€™(0, 0)$, for #16.
7. The coordinates of $\triangle DEF$ are $D(4, -2), E(7, -4)$ and $F(5, 3)$. Translate $\triangle DEF$ to the right 5 units and up 11 units. Write the translation rule.
8. 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.

1. $y = -2x+17$
2. Kite
3. $y = -2x+5$

8 , 9 , 10

Feb 22, 2012

Dec 11, 2014