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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

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.

  1. What is the image of ?
  2. What is the image of ?
  3. What is the image 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?

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.

  1. Find the lengths of all the sides of .
  2. Find the lengths of all the sides of .
  3. What can you say about and ? Can you say this for any translation?
  4. If was the preimage and was the image, write the translation rule for #14.
  5. If was the preimage and was the image, write the translation rule for #15.
  6. Find the translation rule that would move to , for #16.
  7. The coordinates of are and . Translate to the right 5 units and up 11 units. Write the translation rule.
  8. 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. 

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Vocabulary

rigid transformation

A transformation that does not change the size or shape of a figure, also known as an isometry or congruence transformation.

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.

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.

Image Attributions

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