What if Lucy lived in San Francisco, , and her parents lived 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?

### Geometric Translations

A **transformation** is an operation that moves, flips, or changes a figure to create a new figure. A **rigid transformation** is a transformation that preserves size and shape. The rigid transformations are: translations (discussed here), 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**. Rigid transformations are also called **congruence transformations**. 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.”

A **translation** is 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. Another way to write a translation rule is to use vectors. A **vector** is 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 **terminal point**. The terminal point always has the arrow pointing towards it and has the half-arrow over it in the label.

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.

#### Graphing and Translating

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

The translation notation tells us that we are going to move the square to the left 2 and up 3.

#### Naming and Writing Vectors

1. Name the vector and write its component form.

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 .

2. Name the vector and write its component form.

The vector is . The component form of is .

#### Earlier Problem 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.

### Examples

#### Example 1

Find the translation rule for to .

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 .

#### Example 2

Draw the vector with component form .

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

#### Example 3

Triangle has coordinates and . Translate using the vector . Determine the coordinates of .

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

#### Example 4

Write the translation rule for the vector translation from #3.

To write as a translation rule, it would be .

### Review

- 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.

For questions 17-19, name each vector and find its component form.

- 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 .

### Review (Answers)

To view the Review answers, open this PDF file and look for section 12.3.