9.1—Translations

Vocabulary

Figures in a plane can be reflected, rotated, or translated to produce new figures.

The new figure is called the image, and the original figure is called the preimage.

The operation that maps, or moves, the preimage onto the image is called a transformation.

Identifying Transformations

There are three basic transformations: reflections, rotations, and translations.

reflection in a line rotation about a point translation

“flip” “turn” “slide”

Key: Preimage

Image

When you name an image, take the corresponding point of the preimage and add a prime symbol. For instance, if the preimage is A, then the image is A’, read as “A prime”.

Name the transformation that maps the blue pickup truck (preimage) onto the red pickup (image).

translation reflection rotation

Identifying Isometries

An isometry is a transformation that preserves lengths, angle measures, parallel lines, and distances between points.

Also called “rigid transformations”.

Which of the following transformations appear to be isometries?

Reflection Dilation Rotation

Translation is an Isometry

You can describe the transformation in the diagram by writing “ΔABC is mapped onto ΔDEF.” This is a translation (slide)

You can also use arrow notation as follows:

ΔABC → ΔDEF

The order in which the vertices are listed specifies the correspondence.

A → D, B → E, C → F

Finding the Image of a TranslationSketch a triangle with vertices A(-1, -3), B(1, -1), and C(-1, 0). Then sketch the image of the triangle under the translation (x, y)→(x – 3, y + 4).

1. Plot the points of the original triangle.2. Shift each x-coordinate 3

units to the LEFT and each y-coordinate 4 units UP.

*positive - right or up negative - left or down

ΔABC ΔA’B’C’

A(-1, -3) A’(-4, 1)B(1, -1) B’(-2, 3)C(-1, 0) C’(-4, 4)

Rule:

x value: subtract 3 units

y value: add 4 units

y

x

A

BCA’

B’C’

What are the images of the vertices of ΔABC for the translation (x, y)→(x + 3, y – 2)?

Graph vertices of pre-image A(-1, -3), B(1, -1), and C(-1, 0). Then sketch the image of the triangle under the translation (x, y)→(x + 3, y – 2).

1. Plot the points of the original triangle.2. Shift each x-value 3 RIGHT

y- value 2 DOWN.

ΔABC ΔA’B’C’

A(-1, -3) A’(2, -5)B(1, -1) B’(4, -3)C(-1, 0) C’(2, -2)

Rule:

x value: add 3 units

y value: subtract 2 units

y

x

A

BC

A’

B’C’

ΔABC→ΔA’B’C’ by translation is defined by (x, y)→(x – 5, y). The coordinates of the vertices of ΔABC are A(7, 4), B(-1, -1), and C(3, -5). What are the coordinates of the vertices of ΔA’B’C’ ?

A’(2, 4), B’(-6, -1), C’ (-2, -5)

Wrap-Up

• What is a transformation?An operation that moves or maps a figure

• Name three isometric transformations.Reflection, rotation, and translation