# 12.2: Translations

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Graph a point, line, or figure and translate it and 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 and ?
3. Find the equation of the line parallel to #1 that passes through (4, -3).

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

a) The translation rule for to .

b) The translation rule for to .

c) The translation rule for to .

d) The translation rule for to . 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 , then the image would be , said “a prime.” If there is an image of , that would be labeled , 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 .

Example 1: Graph square and . Find the image after the translation . Then, graph and label the image.

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

Example 2: Find the translation rule for to .

Solution: Look at the movement from to . The translation rule is .

Example 3: Show from Example 2.

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

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

Example 4: Triangle has coordinates and . Translate to the left 4 units and up 5 units. Determine the coordinates of .

Solution: Graph . To translate , subtract 4 from each value and add 5 to each value.

The rule would be .

Know What? Revisited

a)

b)

c)

d)

## 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 for questions 1-7.

1. What is the image of ?
2. What is the image of ?
3. What is the image of ?
4. What is the image of ?
5. What is the preimage of ?
6. What is the image of ?
7. Plot and from the questions above. What do you notice?

The vertices of are and . Find the vertices of , given the translation rules below.

In questions 14-17, is the image of . Write the translation rule.

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

1. Find the lengths of all the sides of .
2. Find the lengths of all the sides of .
3. What can you say about and ? Can you say this for any translation?
4. If was the preimage and was the image, write the translation rule for #14.
5. If was the preimage and was the image, write the translation rule for #15.
6. Find the translation rule that would move to , for #16.
7. The coordinates of are and . Translate to the right 5 units and up 11 units. Write the translation rule.
8. The coordinates of quadrilateral are and . Translate to the left 3 units and down 7 units. Write the translation rule.

## Review Queue Answers

1. Kite

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CK.MAT.ENG.SE.1.Geometry-Basic.12.2