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3.1: Symmetry in Polygons

3.1: Symmetry in Polygons. On the first day of school, Mr Vilani gave his 3 rd grade students 5 new words to spell. On each school day after that he gave them 3 new words to spell. In the first 20 days of school, how many new words had the students been given to spell?. Polygons.

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3.1: Symmetry in Polygons

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  1. 3.1: Symmetry in Polygons 3.1: Symmetry in Polygons

  2. On the first day of school, Mr Vilani gave his 3rd grade students 5 new words to spell. On each school day after that he gave them 3 new words to spell. In the first 20 days of school, how many new words had the students been given to spell? 3.1: Symmetry in Polygons

  3. Polygons A figure is a polygon iff it is a plane figure formed from 3 or more segments such that each segment intersects exactly 2 others, one at each endpoint, and no 2 segments with a common endpoint are collinear. The segments are the sides of the polygon and the endpoints are the vertices of the polygon. 3.1: Symmetry in Polygons

  4. Classifying Polygons 3.1: Symmetry in Polygons

  5. Classifying Polygons 3.1: Symmetry in Polygons

  6. Triangles Defn: A triangle is scalene iff it has no congruent sides. A triangle is isosceles iff it has at least 2 congruent sides. A triangle is equilateral iff all 3 sides are congruent. 3.1: Symmetry in Polygons

  7. Equilateral Polygons Defn: A polygon is equilateral iff all of its sides are congruent. 3.1: Symmetry in Polygons

  8. Equiangular Polygons Defn: A polygon is equiangular iff all of its angles are congruent. 3.1: Symmetry in Polygons

  9. Regular Polygons Defn: A polygon is regular iff it is equilateral and equiangular. 3.1: Symmetry in Polygons

  10. Center of a Regular Polygon Defn: The center of a regular polygon is the point that is equidistant from all vertices of the polygon. Center of the regular polygon 3.1: Symmetry in Polygons

  11. Central Angle of a Regular Polygon Defn: A central angle of a regular polygon is an angle whose vertex is the center of the regular polygon and whose sides contain 2 consecutive vertices. Central angle of the regular polygon 3.1: Symmetry in Polygons

  12. The Central Angle of a Regular Polygon The measure, , of a central angle of a regular polygon with n sides is given by the formula: = 360 n 3.1: Symmetry in Polygons

  13. What is the measure of the central angle of a regular octagon? 3.1: Symmetry in Polygons

  14. Reflectional Symmetry Defn: A figure has reflectional symmetry iff its reflected image across a line coincides exactly with the preimage. The line is called an axis of symmetry. 3.1: Symmetry in Polygons

  15. Triangle Symmetry Conjecture An axis of symmetry in a triangle is the perpendicular bisector of the side it intersects, and it passes through the vertex of the angle opposite that side of the triangle. An equilateral triangle has 3 axes of symmetry. A strictly isosceles triangle has 1 axis of symmetry. A scalene triangle has 0 axes of symmetry. 3.1: Symmetry in Polygons

  16. Axis of Symmetry for a Triangle axis of symmetry 3.1: Symmetry in Polygons

  17. By the Triangle Symmetry Property, is the yellow segment an axis of symmetry for the triangle? Why or why not? 3.1: Symmetry in Polygons

  18. By the Triangle Symmetry Property, is the yellow segment an axis of symmetry for the triangle? Why or why not? 3.1: Symmetry in Polygons

  19. Rotational Symmetry • Defn: A figure has rotational symmetry iff it has at least one rotation image not counting rotation images of 0° or multiples of 360° that coincide with the preimage. 3.1: Symmetry in Polygons

  20. Rotational Symmetry 3.1: Symmetry in Polygons

  21. Rotational Symmetry • All geometric figures have 0° (360°) rotational symmetry. If a figure has only 0° (360°) rotational symmetry, it is said to have trivial rotational symmetry. 3.1: Symmetry in Polygons

  22. Rotational Symmetry • n-fold rotational symmetry: A figure that is said to have n-fold rotational symmetry will rotate onto itself n times if the figure is rotated 360°. 3.1: Symmetry in Polygons

  23. Center of Symmetry The center of the rotation which yields the rotational symmetry is also called the center of symmetry. 3.1: Symmetry in Polygons

  24. Describe all symmetries for a regular hexagon. 6-fold rotational symmetry (60°, 120°, 180°, 240°, 300° and 360°=0°) 6 axes of symmetry: the 3 perpendicular bisectors of each pair of opposite sides and the 3 lines containing the center and a vertex. 3.1: Symmetry in Polygons

  25. Assignment • Pages 143- 146, • # 10 – 26 (evens), 28 – 32 (all), 34 – 40 (evens), 46 – 50 (all), 52, 54 3.1: Symmetry in Polygons

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