# 12.2: Translations and Vectors

## Learning Objectives

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

## 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).
- Find the equation of the line perpendicular to #1 that passes through (4, -3).

**Know What?** Lucy currently lives in San Francisco, , and her parents live in Paso Robles, . She will be moving to Ukiah, , in a few weeks. All measurements are in miles. Find:

a) The component form of and .

b) Lucy’s parents are considering moving to Fresno, . Find the component form of and .

c) Is Ukiah or Paso Robles closer to Fresno?

## Transformations

Recall from Lesson 7.6, we learned about dilations, which is a type of transformation. Now, we are going to continue learning about other types of transformations. All of the transformations in this chapter are rigid transformations.

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

**Rigid Transformation:** A transformation that preserves size and shape.

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***. Rigid transformations are also called congruence transformations.**

*isometry*Also in Lesson 7.6, we learned how to label an image. If the preimage is , then the image would be labeled , said “a prime.” If there is an image of , that would be labeled , said “a double prime.”

## Translations

The first of the rigid transformations is a translation.

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

In the coordinate plane, we say that a translation moves a figure units and units.

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

**Solution:** The translation notation tells us that 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 . is (-3, 3) and is (3, -1). The change in is 6 units to the right and the change in is 4 units down. Therefore, the translation rule is .

From both of these examples, we see that a translation preserves congruence. Therefore, ** a translation is an isometry**. We can show that each pair of figures is congruent by using the distance formula.

**Example 3:** Show from Example 2.

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

## Vectors

Another way to write a translation rule is to use vectors.

**Vector:** A quantity that has direction and size.

In the graph below, the line from to , or the distance traveled, is the vector. This vector would be labeled because is the ** initial point** and is the

**. The terminal point always has the arrow pointing towards it and has the half-arrow over it in the label.**

*terminal point*

The ** component form** of combines the horizontal distance traveled and the vertical distance traveled. We write the component form of as because travels 3 units to the right and 7 units up. Notice the brackets are pointed, , not curved.

**Example 4:** Name the vector and write its component form.

a)

b)

**Solution:**

a) The vector is . From the initial point to terminal point , you would move 6 units to the left and 4 units up. The component form of is .

b) The vector is . The component form of is .

**Example 5:** Draw the vector with component form .

**Solution:** The graph above is the vector . From the initial point it moves down 5 units and to the right 2 units.

The positive and negative components of a vector always correlate with the positive and negative parts of the coordinate plane. We can also use vectors to translate an image.

**Example 6:** Triangle has coordinates and . Translate using the vector . Determine the coordinates of .

**Solution:** It would be helpful to graph . To translate , add each component of the vector to each point to find .

**Example 7:** Write the translation rule for the vector translation from Example 6.

**Solution:** To write as a translation rule, it would be .

**Know What? Revisited**

a)

b)

c) You can plug the vector components into the Pythagorean Theorem to find the distances. Paso Robles is closer to Fresno than Ukiah.

## Review Questions

- What is the difference between a vector and a ray?

Use the translation for questions 2-8.

- What is the image of ?
- What is the image of ?
- What is the preimage 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? Write a conjecture.

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

In questions 13-16, is the image of . Write the translation rule.

- Verify that a translation is an isometry using the triangle from #15.
- If was the
*preimage*and was the image, write the translation rule for #16.

For questions 19-21, name each vector and find its component form.

For questions 22-24, plot and correctly label each vector.

- The coordinates of are and . Translate using the vector and find the coordinates of .
- The coordinates of quadrilateral are and . Translate using the vector and find the coordinates of .

For problems 27-29, write the translation rule as a translation vector.

For problems 30-32, write the translation vector as a translation rule.

## Review Queue Answers

- Kite