Ch 9: Transformations

9-1 Translations
9-2 Reflections / 9-3 Rotations
9-4 Symmetry

9-1Translations:

Focused Learning Target: I will be able to
  • Identify Isometries.
  • Find translation images of figures.
/ Standard(s): Geometry 22.0. Students know the effect of rigid motions on figures in the coordinate plane and space, including translations.
Vocabulary:
  • Transformation
  • Preimage
  • Image
/
  • Isometry
  • Rigid motion
  • Translation

Transformation: a change in the position, shape or size of a geometric figure.

Examples:

The original figure (before the transformation) is the preimage. The resulting figure (after the transformation) is the image.

Isometry: a transformation in which the preimage and image are congruent. An isometry can also be thought of as a rigid motion because lengths and angles are preserved. (same shape and size)

Identifying Isometries:

Does the transformation appear to be an a isometry? Explain.

I’ll do one / We’ll do one together / You try

A transformation maps a figure (preimage) onto its image and may be described with arrow notation. Prime ( ‘ ) notation is sometimes used to identify image points. In the diagram below, K’ is the image of K

/ Notice that you name the corresponding parts of the preimage and the image in the same order, as you do for corresponding points of congruent or similar figures.

Naming Images and Corresponding Parts:

I’ll do one:


a. Name the images of
  1. List all pairs of corresponding sides.

We’ll try one:

In the diagram, NIDSUP.
a. Name the images of and point D
b. List all pairs of corresponding sides.

You try:

In the diagram, C’D’E’F’ is the image of CDEF.
a. Name the images of and
b. List all pairs of corresponding sides.

Finding a translation image

I’ll do one:

We’ll do one: / You try:

Writing a rule to describe a Translation:

I’ll do one:

Write a rule to describe the translation PQRSP’Q’R’S’.

We’ll do one: / You try:

9-2Reflections:

Focused Learning Target: I will be able to
  • Find reflection images of figures
/ Standard(s):
Geometry 22.0. Students know the effect of rigid motions on figures in the coordinate plane and space, including reflections.
Vocabulary:
  • reflection

Reflection (flip): an isometry in whicha figure and its image have opposite orientations. It is the same as a “mirror image”.

Example:

To reflect a figure across a line, use the following rules:

  • If point A is on the line, then the image of A is itself. (A = A’).
  • If point B is not on the line, then the line is the perpendicular bisector of . (B and B’ are the same distance from the line, but on opposite sides).

Example:

Finding Reflection images:

I’ll do one:

Find the image of points P, Q and R reflected across the line y=2

We’ll do one together: / You Try:
Find the image of points P, Q and R reflected across the y-axis
/ Find the image of points A, B & C reflected about the line y = 2

Drawing reflection images:

I’ll do one:

Given points A(3,4), B(0,1) & C(2,3), draw and it’s reflection image across the x- axis.

We’ll do one together: / You Try:
Given points J(1,4), A(3,5) & R(2,1), draw and it’s reflection image across the line x= -1
/ Given points A(3,4), B(0,1) & C(2,1), draw and it’s reflection image across the x- axis.

9-3 Rotations:

Focused Learning Target: I will be able to
  • draw and identify rotation images of figures
/ Standard(s):
Geometry 22.0. Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations.
Vocabulary:
  • rotation
/
  • center of rotation
  • center of a regular polygon

Rotation: a rotation is the action performed when an object is “spun” around a specific point. The point used to spin the object is the center of rotation. The center of rotation can be located on, or off the object. In addition, the rotation can be full , or partial.

Examples:

When rotating an image about the origin, in multiples of , the resulting image is found by:

Clockwise: switching the values of x and y and negating the resulting y-coordinate for each turn. (a rotation of would use 2 switches, a rotation of would use 3 switches, etc.)

Counter-clockwise: switching the values of x and y and negating the resulting x-coordinate for each turn. (a rotation of would use 2 switches, a rotation of would use 3 switches, etc.)

Alternative methods:

- (For both clockwise and counterclockwise) switch the values of x and y as stated above, but determine the signs based upon the quadrant the image is in.

- 90o clockwise or 270o counter-clockwise (x, y)  (-y, x)

- 90o counter-clockwise or 270o clockwise (x, y)  (y, -x)

- 180o (for both clockwise or counter-clockwise) (x, y)  (-x, -y)

Drawing a Rotation Image about the origin:

I’ll do one:

Draw the image of for a rotation clockwise about the origin.

We’ll do one together: / You Try:
Draw the image of for a counter-clockwise rotation about (0, 0) / Draw the image of for a rotation about the origin.

Identifying a rotational Image:

I’ll do one: / We’ll do one together:
What point represents a 72º clockwise rotation of point B?
/ What point represents a 135º clockwise rotation of point C?

You try:

What point represents a 240o clockwise rotation of point C?

9-4Symmetry

Focused Learning Target: I will be able to identify the type of symmetry in a figure
Vocabulary:
  • Symmetry
  • Reflectional symmetry
/
  • Line symmetry
  • Rotational symmetry
  • Point symmetry

There are two types of symmetry:

Reflectional symmetry– the figure has a line of symmetry that bisects the figure creating a mirror image (imagine folding a figure along a line so that both halves match perfectly).

Rotational symmetry– the figure can be rotated around its center by 180o or less and still be in the exact same position.

Identifying Reflectional Symmetry:

Is it possible to have more than one line of symmetry?

I’ll do one:
/ We’ll do one together:
/ You try:

Identifying Rotational Symmetry:

Does each object have rotational symmetry? If so, give the angle(s) of rotation.

I’ll do one:
/ We’ll do one together:
/ You try:

Does each object above have reflectional symmetry? If so, draw in the lines of symmetry.

Group work:

Using only capital letters, which ones have reflectional symmetry? Which capital letters have rotational symmetry and by how many degrees?

EX. The letter H has 2 lines of symmetry and a rotational symmetry of 180o

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