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Geometric Translations

Movement of every point in a figure the same distance in the same direction.

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Geometric Translations

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 

  1. What is the difference between a vector and a ray?

Use the translation for questions 2-8.

  1. What is the image of ?
  2. What is the image of ?
  3. What is the preimage of ?
  4. What is the image of ?
  5. What is the preimage of ?
  6. What is the image of ?
  7. 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.

  1. The coordinates of are and . Translate using the vector and find the coordinates of .
  2. 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. 

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Vocabulary

Center of Rotation

In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.

Image

The image is the final appearance of a figure after a transformation operation.

Preimage

The pre-image is the original appearance of a figure in a transformation operation.

Transformation

A transformation moves a figure in some way on the coordinate plane.

Translation

A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.

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