**Transformations** A transformationis an operation that changes some aspect of the geometric figure to produce a new figure. The new figure is called the image, and the original figure is called the pre-image. C C’ Pre-image Image Transformation A A’ B B’

**Congruence Transformations** A congruence transformation, or isometry, is a type of transformation that changes the position of a figure without changing its size or shape. • In other words, in an isometry, the pre-image is congruent to the image. • There are three basic isometries…

**Isometries** Which of the following transformations is not an isometry?

**Tessellations** An interesting application of transformations is a tessellation. A tessellation is a tiling of a plane with one or more shapes with no gaps or overlaps. They can be created using transformations.

**Tessellations**

**Tessellations**

**Vectors** Translations are usually done with a vector, which gives a direction and distance to move our shape.

**Vectors** Translations are usually done with a vector, which gives a direction and distance to move our shape.

**Transformation Coordinate Rules** What are the new coordinates of the point (x, y) under each of the following transformations? • Translation under the vector a, b • Reflection across the x-axis Reflection across the y-axis • Reflection across the line y= x Reflection across the line y = -x • Rotation of 90° around the origin

**Transformation Coordinate Rules** Coordinate Notation for a Translation You can describe a translation of the point (x, y) under the vector a, b by the notation:

**Transformation Coordinate Rules** Coordinate Notation for a Reflection

**Transformation Coordinate Rules** Coordinate Notation for a Rotation

**Example 1** Draw and label ABC after each of the following transformations: • Reflection across the x-axis • Reflection across the y-axis • Translation under the vector -3, 5

**Example 2** What translation vector was used to translate ABC to A’B’C’? Write a coordinate rule for the translation. Vector: a, b = 10, -2 Rule: (x, y) (x + 10, y – 2)

**Example 3** Draw the image of ABC after it has been rotated 90° counterclockwise around the origin.

**Example 3** Draw the image of ABC after it has been rotated 90° counterclockwise around the origin.

**Example 4a** Does the order matter when you perform multiple transformations in a row?

**Example 4b** Does the order matter when you perform multiple transformations in a row?

**Example 4c** Does the order matter when you perform multiple transformations in a row?

**Composition of Transformations** Two or more transformations can be combined to make a single transformation called a composition of transformations.

**Composition of Transformations** When the transformations being composed are of different types (like a translation followed by a reflection), then the order of the transformations is usually important.

**Glide Reflection** A special type of composition of transformations starts with a translation followed by a reflection. This is called a glide reflection.

**Glide Reflection** A special type of composition of transformations starts with a translation followed by a reflection. This is called a glide reflection.