MA 111 Review for Exam 4
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- Alvin Harrell
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1 MA 111 Review for Exam 4 Exam 4 (given in class on Thursday, April 12, 2012) will cover Chapter 11. You should: understand how to carry out each of the following four motions: Reflection Rotation Translation Glide reflection know what it means to be a proper or improper motion, and be able to characterize the four motions as either proper or improper. know what a fixed point is and be able to characterize the four motions by the number of fixed points each one has. know how many point-image pairs are needed to completely determine each motion. given only information about a point(s) and its image(s), be able to construct the other identifying features of a motion. For instance, given one point-image pair for a reflection, be able to find the axis of reflection. be able to identify the symmetry type of a finite object as either D n or Z n. be able to draw examples of finite objects with a given symmetry type. understand the types of symmetries in border patterns and be able to use the chart to determine which of the seven symmetry types a given pattern possesses.
2 Practice Problems Use exercises in the text to supplement these for extra practice. (The odd-numbered problems have solutions in the back of the book so that you can check your answers.) Review the homework exercises, quizzes, and the examples in the text. 1. In each of the following cases, use the properties to decide if the motion is a reflection, a rotation, a translation, or a glide reflection. (a) The motion is improper, and when the same motion is applied twice, we get the identity motion. (b) The motion is proper and has one fixed point. (c) The motion is proper, and when the same motion is applied twice, we get the identity motion. (d) The motion is improper, and when the same rigid motion is applied twice, we get a translation. (e) The motion is proper, and when the same rigid motion is applied twice, we get a translation. 2. For each of the following finite shapes, draw all lines of reflectional symmetry, if any, state the measure of the smallest angle of rotational symmetry, and state its symmetry type. (a) (d) (b) (e) (c) (f)
3 3. Draw an example of a letter which has the symmetry type... (a) D 1 (b) Z 1 (c) Z 2 (d) D 2 4. Explain why a circle has symmetry type D. 5. For the following questions, use the border pattern below: Does it have... (a) horizontal reflectional symmetry? If yes, draw the axis of reflection. (b) vertical reflectional symmetry? If yes, draw the axis of reflection. (c) half-turn symmetry? If yes, mark one rotocenter. (d) translational symmetry? If yes, draw one vector of translation. (e) glide reflectional symmetry? If yes, draw the axis and a vector for the glide reflection. (f) What is its symmetry type?
4 6. Describe what you must do to find the axis of reflection given a point-image pair under the reflection. 7. Describe what you must do to find the rotocenter and angle of rotation given two point-image pairs under the rotation. 8. Describe what you must do to find the vector of translation given a point-image pair under the translation. 9. For the following questions, use the border pattern below: Does it have... (a) horizontal reflectional symmetry? If yes, draw the axis of reflection. (b) vertical reflectional symmetry? If yes, draw the axis of reflection. (c) half-turn symmetry? If yes, mark one rotocenter. (d) translational symmetry? If yes, draw one vector of translation. (e) glide reflectional symmetry? If yes, draw the axis and a vector for the glide reflection. (f) What is its symmetry type?
5 10. Carefully draw a finite shape with symmetry type Z Carefully draw a finite shape with symmetry type D For each of the following finite shapes, draw all lines of reflectional symmetry, if any, state the measure of the smallest angle of rotational symmetry, and state its symmetry type. (a) (d) (b) (e) (c) (f)
6 13. In the following figure, find the image of triangle PQR under the glide reflection with vector v and axis of reflection m. Q P R v m 14. In the following figure, find the image of point P when it is rotated about point Q by an angle of 90 degrees counterclockwise. Label the new point P. P Q
7 15. In the following figure, find the image of the flag when it is reflected over both of the lines shown below. Describe the result as a single rigid motion.
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