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Dilations

For Exercises 2 and 3, XYZ has vertices X (3, 1), Y (2, –4), and Z (–2, 0). For Exercises 4 and 5, DIL is a dilation image of DAT. 2. Use scalar multiplication to find the image of XYZ for a dilation with center (0, 0) and scale factor 2.5. Dilations. Lesson 9-5.

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Dilations

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  1. For Exercises 2 and 3, XYZ has vertices X(3, 1), Y(2, –4), and Z(–2, 0). For Exercises 4 and 5, DIL is a dilation image of DAT. 2. Use scalar multiplication to find the image of XYZ for a dilation with center (0, 0) and scale factor 2.5. Dilations Lesson 9-5 Lesson Quiz 1. A model is a reduction of a real tractor by the scale factor of 1 : 16. Its dimensions are 1.2 ft by 0.6 ft by 0.625 ft. Find the actual dimensions of the tractor. 19.2 ft by 9.6 ft by 10 ft X (7.5, 2.5), Y (5, –10), Z (–5, 0) 3. Draw and label the preimage and image. 4. Identify the center of dilation. 5. Find the scale factor. D 4 9-6

  2. Compositions of Reflections Lesson 9-6 Check Skills You’ll Need (For help, go to Lessons 9-1 and 9-2.) Given points R(–1, 1), S(–4, 3), and T(–2, 5), draw RST and its reflection image in each line. (Just state the coordinates for R’, S’, and T’.) 1. the y-axis 2. the x-axis 3.y = 1 Draw RST described above and its translation image for each translation vector. (Just state the coordinates for R’, S’, and T’.) 4. (x , y)  (x, y– 3) 5. (x , y)  (x + 4, y) 6. (x , y)  (x + 2, y– 5) 7. Copy the figure. Draw the image of the figure for a reflection across DG. Check Skills You’ll Need 9-6

  3. 1. 4. 5. 6. 2. 3. Compositions of Reflections Lesson 9-6 Check Skills You’ll Need Solutions 7. 9-6

  4. Compositions of Reflections Lesson 9-6 9-6

  5. Compositions of Reflections Lesson 9-6 9-6

  6. B B’ B’’ A C C’ A’ A’’ C’’ Reflections and Translations m n A’’B’’C’’ has the same orientation as ABC. Reflecting ABC twice is equal to a translation.

  7. Compound Reflections • If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is a translation. • If P’’ is the image of P, then PP’’ = 2d, where d is the distance between lines k and m.

  8. Reflections and Translations m n d B B’ B’’ A C C’ A’ A’’ C’’ 2d

  9. Compositions of Reflections Lesson 9-6 9-6

  10. Compound Reflections • If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P. k m P

  11. The amount of the rotation is twice the measure of the angle between lines k and m. Compound Reflections k m x 2x P

  12. Compositions of Reflections Lesson 9-6 9-6

  13. Compositions of Reflections Lesson 9-6 Summary 9-6

  14. Compositions of Reflections Lesson 9-6 9-6

  15. Compositions of Reflections Lesson 9-6 If two figures are congruent and have opposite orientations (but are not simply reflections of each other), then there is a slide and a reflection that will map one onto the other. A glide reflection is the composition of a glide (translation) and a reflection across a line parallel to the direction of translation. Line of Reflection 9-6

  16. Compositions of Reflections Lesson 9-6 Additional Examples Recognizing a Transformation 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 9-6

  17. First, find the reflection image in line . It no longer looks like a 4. The arrow is perpendicular to lines and m with length equal to twice the distance from to m. Compositions of Reflections Lesson 9-6 Additional Examples Composition of Reflections Across Parallel Lines Find the image of the figure for a reflection in line and then in line m. 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. Quick Check 9-6

  18. Find the image of D through a reflection in line x. Find the image of the reflection through another reflection in line y. Compositions of Reflections Lesson 9-6 Additional Examples Composition of Reflections in Intersecting Lines The letter D is reflected across line x and then across line y. Describe the resulting rotation. 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 9-6

  19. First, translate ABC by  0, 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) Compositions of Reflections Lesson 9-6 Additional Examples 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. 9-6

  20. Then, reflect the translated image in the line x = 1. A (6, 7), B (–4, 4), and The glide reflection image A B C has vertices C (2, 2). Compositions of Reflections Lesson 9-6 Additional Examples (continued) Quick Check 9-6

  21. Compositions of Reflections Lesson 9-6 Additional Examples Classifying Isometries 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 vertical line. Quick Check 9-6

  22. 4.PQ has endpoints P(4, 15) and Q(–6, 10). Find the image of PQ for a glide reflection where the glide is 0,–8 and the reflection line is x = 0. Compositions of Reflections Lesson 9-6 Lesson Quiz Use the diagram for Exercises 1–3. 1. Find the image of L for a reflection in line a and then in line b. 2. Find the image of L for a reflection in line a and then in line c. 3. Describe the rotation in Exercise 2. 180° rotation with center of rotation at the intersection of lines a and c Check that students’ images have endpoints P (–4, 7) and Q (6, 2). 5. Name the four types of isometries. glide reflection, reflection, rotation, translation 9-6

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