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? After completing this Concept, you'll be able to translate a figure like this one in the coordinate plane.

### Watch This

Transformation: Translation CK-12

### Guidance

*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 (discussed here), reflections (discussed elsewhere), and rotations (discussed elsewhere). 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 .

#### Example A

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 B

Find the translation rule for to .

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

#### Example C

Show from Example B.

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

Since all three pairs of corresponding sides are congruent, the two triangles are congruent by SSS.

Transformation: Translation CK-12

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### Guided Practice

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

Use the translation for questions 2-4.

2. What is the image of ?

3. What is the image of ?

4. What is the image of ?

**Answers:**

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

The rule would be .

2.

3.

4.

### Explore More

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.

### Answers for Explore More Problems

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