Section 12-4 Compositions of Reflections SPI 32D: determine whether the plane figure has been translated given a diagram and vice versa Objectives: Use compositions of reflections Identify glide reflections Recognize a Transformation 72 72 If two figures are , there is a transformation that maps one figure onto the other. Is one figure a reflection of the other? HI HI If there is no reflection, then the figures are either translated or rotated images of each other. Is one figure a reflection of the other?

Express Compositions as Reflections Any translation or rotation can be expressed as two reflections Theorem 12-1 A translation or rotation is a composition of two reflections. Theorem 12-2 A composition of reflections in two parallel lines is a translation. Theorem 12-3 A composition of reflections in two intersecting lines is a rotation.

Composition of Reflections in 2 Parallel Lines (Results in a Translation) Find the image of the figure for a reflection in line ℓ and then in line m. First, find the reflection image in line ℓ . It no longer looks like a 4. Then, find the image of the first reflection in line m. The final image is a translation of the original figure. The arrow shows the direction and distance of the translation. The arrow is perpendicular to lines ℓ and m with length equal to twice the distance from ℓ to m.

Composition of Reflections in Intersecting Lines (Results in Rotation) The letter D is reflected in line x and then in line y. Describe the resulting rotation. Find the image of D through a reflection in line x. Find the image of the reflection through another reflection in line y. The composition of two reflections in intersecting lines is a rotation. The center of rotation is the point where the lines intersect, and the angle is twice the angle formed by the intersecting lines.

Theorem 12-4 Fundamental Theorem of Isometries Glide Reflections Theorem 12-4 Fundamental Theorem of Isometries In a plane, one of two congruent figures can be mapped onto the other by a composition of at most 3 reflections. Composition of a glide (translation) and a reflection in a line parallel to the glide vector

Finding a Glide Reflection Image ∆ABC has vertices A(–4, 5), B(6, 2), and C(0, 0). Find the image of ∆ABC for a glide reflection where the glide is 0, 2 and the reflection line is x = 1. First, translate ABC by 0, 2. (6, 2) (6 + 0, 2 + 2), or (6, 4) (–4, 5) (–4 + 0, 5 + 2), or (–4, 7) (0, 0) (0 + 0, 0 + 2), or (0, 2)

Finding a Glide Reflection Image (continued) Then, reflect the translated image in the line x = 1. The glide reflection image A B C has vertices A (6, 7), B (–4, 4), and C (2, 2).

Classifications of Isometries Theorem 12-5 Isometry Classification There are ONLY four isometries. They are:

Classify Isometries Each figure below is an isometry image of the figure at the right. 1. Tell whether their orientations are the same or opposite. 2. Classify the isometry as either a: Reflection Glide Reflection Translation Rotation