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Geometric Translations

Movement of every point in a figure the same distance in the same direction.

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Geometric Translations

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? After completing this Concept, you'll be able to translate a figure like this one in the coordinate plane.

Guidance

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 (discussed here), Transformation: Reflection , and Transformation: Rotation . 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 $(x,y) \rightarrow (x+5,y+3)$ .

Example A

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.

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

$(x,y) &\rightarrow (x-2,y+3)\\S(1,2) &\rightarrow S'(-1,5)\\Q(4,1) &\rightarrow Q'(2,4)\\R(5,4) &\rightarrow R'(3,7)\\E(2,5) &\rightarrow E'(0,8)$

Example B

Find the translation rule for $\triangle TRI$ to $\triangle T'R'I'$ .

Look at the movement from $T$ to $T'$ . The translation rule is $(x,y) \rightarrow (x+6,y-4)$ .

Example C

Show $\triangle TRI \cong \triangle T'R'I'$ from Example B.

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

$& \underline{\triangle TRI} && \underline{\triangle T'R'I'}\\& TR=\sqrt{(-3-2)^2+(3-6)^2}=\sqrt{34} && T'R'=\sqrt{(3-8)^2+(-1-2)^2}=\sqrt{34}\\& RI=\sqrt{(2-(-2))^2+(6-8)^2}=\sqrt{20} && R'I'=\sqrt{(8-4)^2+(2-4)^2}=\sqrt{20}\\& TI=\sqrt{(-3-(-2))^2+(3-8)^2}=\sqrt{26} && T'I'=\sqrt{(3-4)^2+(-1-4)^2}=\sqrt{26}$

Since all three pairs of corresponding sides are congruent, the two triangles are congruent by SSS.

Guided Practice

1. 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 $\triangle A'B'C'$ .

Use the translation $(x,y) \rightarrow (x+2,y-5)$ for questions 2-4.

2. What is the image of $A(-6, 3)$ ?

3. What is the image of $B(4, 8)$ ?

4. What is the image of $C(5, -3)$ ?

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

$& A(3,-1) \rightarrow (3-4,-1+5)=A'(-1,4)\\& B(7,-5) \rightarrow (7-4,-5+5)=B'(3,0)\\& C(-2,-2) \rightarrow (-2-4,-2+5)=C'(-6,3)$

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

2. $A' (-4, -2)$

3. $B' (6, 3)$

4. $C' (7, -8)$

Practice

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 $\triangle ABC$ are $A(-6, -7), B(-3, -10)$ and $C(-5, 2)$ . Find the vertices of $\triangle A'B'C'$ , 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, $\triangle A'B'C'$ 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 $\triangle A'B'C'$ .
3. What can you say about $\triangle ABC$ and $\triangle A'B'C'$ ? Can you say this for any translation?
4. If $\triangle A'B'C'$ was the preimage and $\triangle ABC$ was the image, write the translation rule for #14.
5. If $\triangle A'B'C'$ 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 $A'(0, 0)$ , 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.

Vocabulary Language: English Spanish

rigid transformation

rigid transformation

A transformation that does not change the size or shape of a figure, also known as an isometry or congruence transformation.
Transformation

Transformation

A transformation moves a figure in some way on the coordinate plane.
Translation

Translation

A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.
Center of Rotation

Center of Rotation

In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.
Image

Image

The image is the final appearance of a figure after a transformation operation.
Preimage

Preimage

The pre-image is the original appearance of a figure in a transformation operation.