### Translations

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**, reflections, and rotations. The new figure created by a transformation is called the **image**. The original figure is called the **preimage**. 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.”

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 .

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?

### Examples

#### Example 1

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

Graph . To translate , subtract 4 from each value and add 5 to each value of its coordinates.

The rule would be .

#### Example 2

Using the translation , what is the image of ?

#### Example 3

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

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

#### Example 4

Find the translation rule for to .

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

### Review

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 (Answers)

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