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Symmetry

1. point A ; a 90° rotation about the center 2. point H ; a 180° rotation about the center 3. AB ; a reflection in AE 4. GH ; a reflection in AE. Symmetry. Lesson 9-4. Check Skills You’ll Need. (For help, go to Lessons 9-1 and 9-3.).

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Symmetry

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  1. 1. point A; a 90° rotation about the center 2. point H; a 180° rotation about the center 3. AB; a reflection in AE 4. GH; a reflection in AE Symmetry Lesson 9-4 Check Skills You’ll Need (For help, go to Lessons 9-1 and 9-3.) The regular octagon at the right is divided into eight congruent triangles. Find the image of the given point or segment for the given rotation or reflection. Check Skills You’ll Need 9-4

  2. 1. There are 8 congruent central angles in the octagon, so each one measures , or 45. Two central angles is 90°, so the image of A is two positions in a counterclockwise direction at point G. 4. The image of GH lies on the opposite side of AE such that every point in GH is the same distance from AE as a corresponding point in the image. One such point that is the same distance from AE as H is B. Another such point that is the same distance from AE as G is C. So the image is mapped onto BC. 360 8 2. From Exercise 1, each central angle measures 45. Since = 4, the image of H is 4 positions in a counterclockwise direction at point D. 180 45 3. The image of AB lies on the opposite side of AE such that every point in AB is the same distance from AE as a corresponding point in the image. One such point that is the same distance from AE as B is H. Also, since A lies on AE, its image also lies on AE. So the image is mapped onto AH. Symmetry Lesson 9-4 Check Skills You’ll Need Solutions 9-4

  3. 90 CW (x, y)  (y, x) 90 CCW (x, y)  (y, x) 180 (x, y)  (x, y) Rotating through an angle other than 90 or 180 requires much more complicated math. Rotation Formulas about the Origin

  4. Symmetry Lesson 9-4 A figure has symmetry if there is an isometry that maps the figure onto itself. 9-4

  5. Symmetry Lesson 9-4 If the isometry is the reflection of a plane figure, the figure has reflectional symmetry or line symmetry. 9-4

  6. Symmetry Lesson 9-4 A figure that has rotational symmetry is its own image for some rotation of 180 degrees or less. 9-4

  7. Symmetry Lesson 9-4 A figure that has point symmetry has 180 degree rotational symmetry. 9-4

  8. Symmetry Lesson 9-4 Additional Examples Identifying Lines of Symmetry Draw all lines of symmetry for an isosceles trapezoid. Draw an isosceles trapezoid. Then draw any lines that divide the isosceles trapezoid so that half of the figure is a mirror image of the other half. There is one line of symmetry. Quick Check 9-4

  9. Symmetry Lesson 9-4 Additional Examples Identifying Rotational Symmetry Judging from appearance, do the letters V and H have rotational symmetry? If so, give an angle of rotation. The letter V does not have rotational symmetry because it must be rotated 360° before it is its own image. The letter H is its own image after one half-turn, so it has rotational symmetry with a 180° angle of rotation. Quick Check 9-4

  10. The nut has 4 lines of symmetry. Symmetry Lesson 9-4 Additional Examples Real-World Connection A nut holds a bolt in place. Some nuts have square faces, like the top view shown below. Tell whether the nut has rotational symmetry about a line and/or reflectional symmetry in a plane. The nut has a square outline with a circular opening. The square and circle are concentric. The nut is its own image after one quarter-turn, so it has 90° rotational symmetry. Quick Check 9-4

  11. 72° rotational symmetry Symmetry Lesson 9-4 Lesson Quiz Tell what type(s) of symmetry each figure has. 1.D 2.O reflectional: horizontal line of symmetry reflectional: horizontal and vertical lines of symmetry; rotational: point symmetry Draw each figure and all its lines of symmetry. 3. isosceles right triangle 4. rhombus that is not a square 5. The star below appears on the United States flag. If the star has line symmetry, sketch it and draw the line(s) of symmetry. If it has rotational symmetry, state the angle of rotation. 9-4

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