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Transcript
CC Geometry R Aim #18: What types of symmetry does a figure have? Do Now: In the diagram below: Sketch Triangle B, the reflection of Triangle A across line x. Sketch Triangle C, the reflection of Triangle B across line y. Complete: Line x and line y are each a _______ __ ________________. y A student did this also, looked at Triangle C and said, "That's not the image of triangle A after two reflections - that's the image of triangle A after a __________________!!" (He was right!) A X When a figure is reflected across a line, and the resulting image is reflected across a second line that intersects the first line, the final image is a ______________ of the original figure. A figure has symmetry if there exists a rigid motion (reflection, rotation) that maps the figure back onto itself. LINE SYMMETRY OF A FIGURE A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line, called the line of symmetry. Every point on one side of the line has a corresponding point on the other side of the line, and the line is equidistant from the corresponding pairs of points. Exercise 1: Draw in all lines of symmetry for each object. C parallelogram rectangle equilateral Δ square isosceles Δ Exercise 2: Construct a line of symmetry for isosceles ΔABC. (AB = CB) B A C Regular Polygon: A polygon is regular if all sides have equal length (is equilateral) and all angles have equal measure (is equiangular). Exercise 3: Construct one line of symmetry that does not intersect a vertex. Then sketch in all remaining lines of symmetry. regular hexagon ROTATIONAL AND IDENTITY SYMMETRY OF A FIGURE A figure has rotational symmetry if the figure can be mapped onto itself by a rotation (turn) between 0˚and 360˚ about the center of the figure, called the center of rotation. Exercise 4: Determine the figures below that have rotational symmetry, and state all the angles that produce rotational symmetry. square rectangle regular pentagon right triangle Exercise 5:Construct and label D, the center of rotation for each regular polygon. Exercise 6: In the equilateral triangle to the right, 0 with circumcenter D, ______ rotations of _____ Exercise 7: ABCD is a square. D A B D C a) Draw all lines of symmetry. Label the center of rotation, S. b) What kinds of symmetry does the square have? c) Name all angles of rotational symmetry. Exercise 8: The figure is a regular octagon. a) How many lines of symmetry are there? b) Name all the angles of rotational symmetry. Exercise 9: What is the minimum number of degrees that maps a) a regular hexagon onto itself? b) a regular octagon onto itself? c) a regular polygon with 10 d) a regular polygon with n sides onto itself? sides onto itself? Exercise 10: James says that the figure has only line symmetry. Jewel says that the figure has only rotational symmetry. Is either of them correct? Explain. Exercise 11: Draw a figure that has rotational symmetry, but not line symmetry, not yet in this Aim. Let's sum it up!! • A line of symmetry divides a figure into two congruent halves that are reflections of each other. • If a figure has rotational symmetry, it can be rotated about the center of 0 0 rotation between 0 and 360 to perfectly overlap itself. • The minimum angle of rotation that makes a regular polygon identical to the original polygon is 360/n (n = number of sides). • If a figure has identity symmetry, each point is mapped onto itself.