# 12.2: Translations

## Learning Objectives

- Graph a point, line, or figure and translate it and units.
- Write a translation rule.

## Review Queue

- Find the equation of the line that contains (9, -1) and (5, 7).
- What type of quadrilateral is formed by and ?
- 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

**. Another word for a rigid transformation is an**

*preimage***or**

*isometry***.**

*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.

- What is the image of ?
- What is the image of ?
- What is the image of ?
- What is the image of ?
- What is the preimage of ?
- What is the image of ?
- 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.

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

## Review Queue Answers

- Kite