3-3. When Kenji spun the flag shown at right very quickly about its pole, he noticed that a threedimensional

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1 Sec Reflections & Rotations and Translations Visualizing, the act of picturing something in your mind, is also helpful when working with shapes. In order to investigate and describe a geometric concept, it is useful to first visualize a shape or action. Reflections can create many beautiful and interesting shapes and can help you learn more about the characteristics of other shapes. Today you will be visualizing in a variety of ways and will develop the ability to find reflections. As you work today, keep the following focus questions in mind: How do we see it? How can we verify our answer? How can we describe it? 3-2. Have you ever noticed what happens when you look in a mirror? Have you ever tried to read words while looking in a mirror? What happens? Discuss this with your team. Then write the following words as they would look if you held this book up to a mirror. What do you notice? a. GEO b. STAR c. WOW 3-3. When Kenji spun the flag shown at right very quickly about its pole, he noticed that a threedimensional shape emerged. a. What shape did he see? Draw a picture of the three-dimensional shape on your paper and be prepared to defend your answer. b. What would the flag need to look like so that a sphere (the shape of a basketball) is formed when the flag is spun about its pole? Draw an example. c. Hunter did not spin the triangular flag all the way around its pole. He only turned it 180. On his paper he recorded the resulting flag image, rather than what he saw while it was moving. He wrote, The flag seems to have flipped over the pole. Which way is the flag pointing now? 1

2 3-4. REFLECTIONS. The shape created by Hunter in problem 3-3 was the result of reflecting the figure over a pole. A reflection across a line is shown in the diagram at right. The reflected figure is called the image of the original figure. a. Why do you think the image is called a reflection? How is the image different from the original? b. Use your visualization skills to predict the reflection of each polygon across the given line of reflection. Then draw the image of the original polygon Sometimes, a motion appears to be a reflection when it really is not. How can you tell if a motion is a reflection? Consider each pair of objects below. Which diagrams represent reflections across the given lines of reflection? Study each situation carefully and be ready to explain your thinking

3 CONNECTIONS WITH ALGEBRA. What other ways can you use reflections? Consider how to reflect a graph as you answer the questions below. a. Graph ΔGLM with vertices G(1, 3), L(2, 7), M(5, 6), and the line y = x. b. Now reflect the triangle over the line y = x. What do you observe? What happens to the x- and y-coordinates of the vertices? c. How does your answer to part (b) relate to the equation of the line of reflection? TRANSLATIONS. Sliding a shape from its original position to a new position is called a translation. For example, the ice cream cone at right has been translated. Notice that the image of the ice cream cone has the same orientation as the original. That is, it is not turned or flipped. 3

4 3-15. ROTATIONS. Flipping a shape about a point from its original position to a new position is called a rotation. For example, the diagram at right shows the result when an ice cream cone is rotated about a point. The rotation of ice cream is an example of a 45 clockwise rotation. The term clockwise refers to a rotation that follows the direction of the hands of a clock, namely. A rotation in the opposite direction ( ) is called counterclockwise Amanda label the vertices in her image square so it is easy to tell which vertices correspond to the vertices of original square. The symbol is read as prime, and the image of the Amanda s square will be called, A prime B prime C prime D prime, written as A B C D, where A is the image of A, B is the image of B, C is the image of C and D is the image of D. a. The diagram at right shows an image of ABCD. Look carefully at the correspondence between the vertices. Can you rotate or reflect the original square to make the letters correspond as shown? If you can reflect, where would the line of reflection be? If you can rotate, where would the point of rotation be? b. This time, Amanda rotated ABCD by 180 about the point as shown. Copy the diagram (both squares and the point) and label the vertices of the image square on the right. If you have trouble, try using tracing paper. 4

5 3-17. Reflections, rotations, and translations of figures are called rigid transformations. Rigid transformations of polygons do not change any of the measures of the angles, or any of the side lengths, in the image. a. The words transformation and translation sound alike and can easily be confused. Discuss with your partner what these words mean and how they are related to each other. Jot down what both came up with. b. Examine ABC and ΔA B C in the graph at right. With your partner, describe at least two different ways to transform ΔABC onto its image ΔA B C. c. Are there always multiple ways to describe any transformation? Discuss this question with your partner and be prepared to share your reasons with the class Consider what you have learned about rigid transformations. a. Would a series of rigid transformations preserve the area of a polygon? That is, would the area of the image always have the same area as the original polygon? Why or why not? 5

6 b. If two polygons have the same area, are they always the image of a series of rigid transformations? HOMEWORK Determine which transformation was used on each pair of polygons below. Some may have undergone more than one transformation, but try to name a single transformation, if possible. Homework Help a. b. c. d. e. f. 6

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