8.5 Transformational Geometry, Symmetry, and Tessellations 519 c) Is it possible to insert a point P within the figure so the figure has 90° rotational symmetry about point P? Explain your answer. No. Any 90° rotation will result in the figure being in a different position than the starting position. d) Is it possible to insert a point P within the figure so the figure has 180° rotational symmetry about point P? Explain your answer. No. Any 180° rotation will result in the figure being in a different position than the starting position. Challenge Problems/Group Activities 49. Tessellation with a Square Create a unique tessellation from a square piece of cardboard by using the method described on page 514. Be creative, using color and sketches to complete your tessellation. Answers will vary. 50. Tessellation with a Hexagon Using the method described on page 514, create a unique tessellation using a regular hexagon like the one shown below. Be creative, using color and sketches to complete your tessellation. Answers will vary. 51. Tessellation with an Octagon? Trace the regular octagon, shown below, onto a separate piece of paper. Try to create a regular tessellation by tracing this octagon repeatedly. Attempt to cover the entire piece of paper where no two octagons overlap each other. What conclusion can you draw about using a regular octagon as a tessellating shape? A regular octagon cannot be used as a tessellating shape. 52. Tessellation with a Pentagon? Repeat Exercise 51 using the regular pentagon below instead of a regular octagon. * Recreational Mathematics 53. Reflective Symmetry Examine each capital letter in the alphabet and determine which letters have reflective symmetry about a horizontal line through the center of the letter. * *See Instructor Answer Appendix b) Other than vertex labels, is the position of square A BC D ′ ′ ′ ′ identical to the position of square ABCD? Yes c) Does square ABCD have 90° rotational symmetry about point P? Yes d) Now rotate the square in the original position, square ABCD, shown above, 180° about point P. Label the rotated square A B C D . ″ ″ ″ ″ * e) Other than vertex labels, is the position of square A B C D ″ ″ ″ ″ identical to the position of square ABCD? Yes f) Does square ABCD have 180° rotational symmetry about point P? Yes 47. Consider the following figure. D C L K E F I J B A G H a) Insert a vertical line m through the figure so the figure has reflective symmetry about line m. * b) Insert a horizontal line l through the figure so the figure has reflective symmetry about line l. * c) Insert a point P within the figure so the figure has 180° rotational symmetry about point P. * d) Is it possible to insert a point P within the figure so the figure has 90° rotational symmetry about point P? Explain your answer. No. Any 90° rotation will result in the figure being in a different position than the starting position. 48. Consider the following figure. D C E F B A G H a) Insert a vertical line m through the figure so the figure has reflective symmetry about line m. * b) Is it possible to insert a horizontal line l through the figure so the figure has reflective symmetry about line l? Explain your answer. *
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