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.
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
Specifically, rigid transformations (or congruence transformations) transform a figure without changing its size or shape.
The three types of rigid transformations are:
To distinguish between the preimage and the image, primes are used to label the image. The arrow \(\rightarrow\) is used to describe a transformation
Another way to describe a reflection is a “flip.”
x-coordinates stay the same while y-coordinates change
x-coordinates change while y-coordinates stay constant
Any line can be a line of reflection, but the lines y = x and y = -x are special cases.
Same as a “reflection in the origin,” a figure is spun around the origin.
Rotation of 90°
Rotation of 180°
Rotation of 270°
Vectors can be used to represent a translation.
The translation rule for this figure is: (x, y) \(\rightarrow\) (x + 6, y + 4).
A dilation is a non-rigid transformation that preserves shape but not size.
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.