Exploration #1: What single transformation is equivalent to a composition of reflections over parallel lines?
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1 Name Geometry Period 1-5 Notes Date Today s Learning Goal: What is special about a composition of reflections over 2 parallel lines? What are special characteristics of a glide reflection? Exploration #1: What single transformation is equivalent to a composition of reflections over parallel lines? Let s continue our work with compositions. Work with your partner and answer the following questions. Be prepared to share your thinking! Task: Perform the following composition on XYZ: r!!!! ο r!!! Pre- Image Image Discover the Relationship! 1. What is special about the two lines of reflection? 2. Compare the final image with the pre- image. How are they related? 3. Can you describe a single transformation that will map the original triangle onto its final image? What would it be? Predict: 4. What do you think might happen when we perform a composition of reflections across two lines that intersect? TOGETHER! Reflections in Parallel Lines Theorem: A composition of reflections over two parallel lines is the same as one single. Name Date
2 Class Discussion: The diagram below shows ABC after a reflection across the x axis, then a translation T!,!. a) Draw a vector showing the translation. b) What is the slope of this vector? What is the slope of the line of reflection ( x axis)? What do you notice about these slopes? The example above is a Glide Reflection. A glide reflection is a composition of a and a. In glide reflections ONLY, the line of reflection will be to the vector of the translation. Fun Fact! Only in Glide Reflections, the order does not matter! How would you be able to check mathematically? Try one! The vertices of PQR are P(2, 1), Q(4, 1), and R(4, 3). Find P Q R, the image of PQR under T 0,- 5 r y- axis. Is the composition you performed a glide reflection? How do you know? Justify your answer!
3 Special Compositions Quick Facts : 1. A composition of reflections over two parallel lines, can also be done as 2. To justify that a composition is a glide reflection you should: 3. Glide reflections are special types of compositions because: Self-Assess! How are you feeling? I Get it! Give me a quiz today! I was okay when we did it as a group but would like guided practice! I feel confused on how to work through these problems or how to set them up on my own.
4 Name: Geometry pd. 1-5 Practice Date: Directions: Complete each of the following special composition problems: Show your work by stating the coordinates of the pre- images, intermediate images and final images for each example and graph final image. Graph: Show Work Here: 1. r (!!!!) r (!!!) Can you describe a single transformation that will map the original triangle onto its final image? 2. Which of these is equivalent to a translation? a. a reflection across one line b. a composition of two reflections across intersecting lines c. a composition of two reflections across parallel lines 3. A figure is reflected across the line y = 2, then reflected across the line y = 4. Which single transformation results in the same image? a. a reflection across the line y = 3 b. a reflection across the line y = 6 c. a translation 2 units up d. a translation 4 units up
5 4. Given diagram below, with k m. A translation maps on to which triangle? Explain your answer! 5. Determine whether the composition is a glide reflection. Justify your answer. Use of the grid is optional. Use triangle A(3, 2), B(5, 7), C(7, 3), as your pre- image. 6. Mathematical Connections! Point B (1,4) is the image of B(3,2) after a reflection over line c. Write an equation for line c. Hint: think back to what you already know/remember about equations of lines! Sketch to help!
6 1-5 Practice Hints 1. Which transformation should you work on first? Can you describe a single transformation that will map the original triangle onto its final image? Look at first figure and last- What transformation happened? 3. Y= lines are horizontal! 4. Translations are glides! 5. Which transformation should you work on first? Organize!
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