Warm-Up Which of the following does not belong?

4.8 Congruence Transformations Intro to Transformations 4.8 Congruence Transformations Objectives: To define transformations To view tessellations as an application of transformations To perform transformations in the coordinate plane using coordinate notation

Transformations A transformation is 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. Pre-image A B C Image A’ B’ C’ Transformation

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

Example 1 Frank is looking to impress his wife by retiling the guest bathroom. Which of the following shapes could he not use to tile the floor?

Example 1 Frank is looking to impress his wife by retiling the guest bathroom. Which of the following shapes could he not use to tile the floor?

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.

Investigation 1 For the following Investigation, we will discover properties of transformations on the coordinate plane. Then we will write a coordinate rule for each transformation.

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° counterclockwise around the origin

Example 2 Transformation Coordinate Notation Image of (8, −13) Translation under vector <−2, 6> Reflection across x-axis Reflection across y-axis Reflection across y = x Reflection across y = −x Rotation 90° CC around origin Rotation 180° CC around origin Rotation 270° CC around origin Rotation 360° CC around 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 3 Draw and label ΔABC after each of the following transformations: Reflection across the x-axis Reflection across the y-axis Translation under the vector <−3, 5>

Example 4 What translation vector was used to translate ABC to A’B’C’? Write a coordinate rule for the translation.

Example 5 Draw the image of ABC after it has been rotated 90° counterclockwise around the origin.

Example 5 Draw the image of ABC after it has been rotated 90° counterclockwise around the origin.

Example 6a Does the order matter when you perform multiple transformations in a row? Translation under <2, −3> Translation under <−4, −1>

Example 6b Does the order matter when you perform multiple transformations in a row? Reflection across y-axis Reflection across x-axis

Example 6c Does the order matter when you perform multiple transformations in a row? Translation under <2, −3> Reflection across y-axis

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.

Example 7 Draw and label ΔABC after the following glide reflection: Translation under the vector <4, −2> Reflection across the line y = x