"9 6 Compositions of Reflections"
Compositions of Reflections LESSON 9-6 Additional Examples Judging by appearances, is one figure a translation image or rotation image of the other? Explain. The figures appear to be congruent, and their orientations are the same. Corresponding sides of the figures appear to be parallel. This suggests that one figure is a translation image of the other and not a rotation image. Quick Check HELP GEOMETRY Compositions of Reflections LESSON 9-6 Additional Examples Find the image of the figure for a reflection across line and then across line m. First, find the reflection image in line . 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 and m with length equal to twice the distance from to m. Quick Check HELP GEOMETRY Compositions of Reflections LESSON 9-6 Additional Examples The letter D is reflected across line x and then across 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. So the letter D is rotated 86° clockwise, or 274° counterclockwise, with the center of rotation at point A. Quick Check HELP GEOMETRY Compositions of Reflections LESSON 9-6 Additional Examples ABC has vertices A(–4, 5), B(6, 2), and C(0, 0). Find the image of ABC for a glide reflection where the translation is (x, y) (x, y + 2) and the reflection line is x = 1. First, translate ABC by the rule (x, y) (x, y + 2). (–4, 5) (–4 + 0, 5 + 2), or (–4, 7) (6, 2) (6 + 0, 2 + 2), or (6, 4) (0, 0) (0 + 0, 0 + 2), or (0, 2) HELP GEOMETRY Compositions of Reflections LESSON 9-6 Additional Examples (continued) Then, reflect the translated image across the line x = 1. The glide reflection image A B C has vertices A (6, 7), B (–4, 4), and C (2, 2). Quick Check HELP GEOMETRY Compositions of Reflections LESSON 9-6 Additional Examples Tell whether orientations are the same or opposite. Then classify the isometry of the letter N. If you turn the figure on the left, it has the same orientation as N. It is a rotation of N. The segment connecting the vertical segments of the letter N slopes down from the left to the right. The segment connecting the vertical segments of the figure on the right slopes up from left to right. So the figure on the right and N have opposite orientations. It is a reflection of N across a horizontal line. Quick Check HELP GEOMETRY