Geometry

Types of Transformations

Big Picture

Transformations move and modify geometric shapes. There are several types of transformations that all transform figures in different ways. These transformations can be rigid (isometric) or non-rigid.

Key Terms

Transformation: An operation that moves, flips, or changes a figure to create a new figure.

Rigid: A transformation that preserves size and shape.

Isometry: Another word for rigid transformation, a transformation that does not change the shape or size of a figure.

Non-Rigid: A transformation that does not preserve size

Preimage: The original figure before a transformation.

Image: The figure after a transformation.

Vector: A quantity that has direction and size

Rigid Transformations

Specifically, rigid transformations (or congruence transformations) transform a figure without changing its size or shape.

The three types of rigid transformations are:

  • Reflection: A transformation that turns a figure into its mirror image by flipping it over a line.
  • Rotation: A transformation that turns a figure around a fixed point to create an image.
  • Translation: A transformation that moves every point in a figure the same distance in the same direction.

To distinguish between the preimage and the image, primes are used to label the image. The arrow \(\rightarrow\) is used to describe a transformation

  • Example: If the preimage is A, then the image would be \(A’\).  \(A \rightarrow A’\)

Reflections

Another way to describe a reflection is a “flip.”

  • Image is the same distance away from the reflection line as the preimage.

Reflection Over Horizontal Lines

x-coordinates stay the same while y-coordinates change

  • If (x, y) is reflected over the x-axis, then the image is (x, -y).
  • If (x, y) is reflected over y = b, then the image is (x, 2b - y).
Reflection Over Horizontal Lines

Reflection Over Vertical Lines

x-coordinates change while y-coordinates stay constant

  • If (x, y) is reflected over the y-axis, then the image is (-x, y).
  • If (x, y) is reflected over y = b, then the image is (x, 2b - y).
Reflection Over Vertical Lines

Reflections over y=x and y=-x

Any line can be a line of reflection, but the lines y = x and y = -x are special cases.

  • If (x, y) is reflected over the y = x, then the image is (y, x).
  • If (x, y) is reflected over y = -x, then the image is (-x, -y).

Geometry

Types of Transformations Cont.

Rotations

Same as a “reflection in the origin,” a figure is spun around the origin.

Rotations

Rotation of 90°

  • If (x, y) is rotated 90° around the origin, then the image will be (-y, x).

Rotation of 180°

  • If (x, y) is rotated 180° around the origin, then the image will be (-x, -y).

Rotation of 270°

  • If (x, y) is rotated 270° around the origin, then the image will be (y, -x).
Rotations

Translations

Vectors can be used to represent a translation.

  • A vector has a length and a direction.
  • A vector going from point A to B is labeled \(\overrightarrow{AB}\). A is the initial point and B is the terminal point. The terminal point has the arrow pointing towards it.
  • The component form of the vector combines the horizontal and vertical distances traveled.
  • In the diagram above, the component form of \(\overrightarrow{AB}\) is <3, 7>.

Translations in the Coordinate Plane

A translation can be described by the notation: \((x, y) \rightarrow (x + a, y + b)\), where the point (x, y) is translated horizontally a units and vertically b units.

The translation rule for this figure is: (x, y)  \(\rightarrow\) (x + 6, y + 4).

  • The vector for this translation is <6, -4>.
TranslationsTranslations in the Coordinate Plane

Dilations

A dilation is a non-rigid transformation that preserves shape but not size.

  • Has a center and scale factor
  • A center is the point of reference for the dilation
  • The scale factor relates how much a figure stretches or shrinks
  • Enlargement or expansion: scale factor > 1
  • Reduction or contraction: 0 < scale factor < 1
  • Always produce a similar shape to the original

In a coordinate plane, a non-rigid transformation transforms \((x, y) \rightarrow (kx, ky)\), where k is the scale factor. If k < 0, then the preimage is rotated around 180° about the center of dilation.

Dilations