3.1: Symmetry in Polygons
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04/21/2304/21/23 3.1: Symmetry in Polygons3.1: Symmetry in Polygons
3.1: Symmetry in 3.1: Symmetry in PolygonsPolygons
04/21/23 3.1: Symmetry in Polygons
On the first day of school, Mr Vilani On the first day of school, Mr Vilani gave his 3gave his 3rdrd grade students 5 new grade students 5 new words to spell. On each school day words to spell. On each school day after that he gave them 3 new after that he gave them 3 new words to spell. In the first 20 days words to spell. In the first 20 days of school, how many new words of school, how many new words had the students been given to had the students been given to spell?spell?
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PolygonsPolygons
A figure is a A figure is a polygonpolygon iff it is a plane iff it is a plane figure formed from 3 or more figure formed from 3 or more segments such that each segment segments such that each segment intersects exactly 2 others, one at intersects exactly 2 others, one at each endpoint, and no 2 segments each endpoint, and no 2 segments with a common endpoint are collinear.with a common endpoint are collinear.
The segments are the The segments are the sides of the sides of the polygonpolygon and the endpoints are the and the endpoints are the vertices of the polygonvertices of the polygon..
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Classifying PolygonsClassifying Polygons
Number of SidesNumber of Sides PolygonPolygon
33 TriangleTriangle
44 QuadrilateralQuadrilateral
55 PentagonPentagon
66 HexagonHexagon
77 HeptagonHeptagon
88 OctagonOctagon
99 NonagonNonagon
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Classifying PolygonsClassifying Polygons
Number of SidesNumber of Sides PolygonPolygon
1010 DecagonDecagon
1111 11-gon11-gon
1212 DodecagonDodecagon
1313 13-gon13-gon
nn n-gonn-gon
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TrianglesTriangles
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.
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Equilateral PolygonsEquilateral Polygons
Defn: A polygon is Defn: A polygon is equilateralequilateral iff all iff all of its sides are congruent.of its sides are congruent.
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Equiangular PolygonsEquiangular Polygons
Defn: A polygon is Defn: A polygon is equiangularequiangular iff all iff all of its angles are congruent.of its angles are congruent.
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Regular PolygonsRegular Polygons
Defn: A polygon is Defn: A polygon is regularregular iff it is iff it is equilateral and equiangular.equilateral and equiangular.
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Center of a Regular PolygonCenter of a Regular Polygon
Defn: The Defn: The center of a regular center of a regular polygonpolygon is the point that is is the point that is equidistant from all vertices of the equidistant from all vertices of the polygon.polygon.
Center of the regular polygon
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Central Angle of a Regular Central Angle of a Regular PolygonPolygon
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
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The Central Angle of a Regular The Central Angle of a Regular PolygonPolygon
The measure, The measure, , of a central angle , of a central angle of a regular polygon with n sides is of a regular polygon with n sides is given by the formula: given by the formula:
==360n
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What is the measure of the central What is the measure of the central angle of a regular octagon?angle of a regular octagon?
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Reflectional SymmetryReflectional Symmetry
Defn: A figure has Defn: A figure has reflectional reflectional symmetrysymmetry iff its reflected image across iff its reflected image across a line coincides exactly with the a line coincides exactly with the preimage. The line is called an preimage. The line is called an axis of axis of symmetrysymmetry..
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Triangle Symmetry Triangle Symmetry ConjectureConjecture
An axis of symmetry in a triangle is An axis of symmetry in a triangle is the perpendicular bisector of the the perpendicular bisector of the side it intersects, and it passes side it intersects, and it passes through the vertex of the angle through the vertex of the angle opposite that side of the triangle.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.
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Axis of Symmetry for a Axis of Symmetry for a TriangleTriangle
axis of symmetry
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By the Triangle Symmetry Property, is the yellow segment an axis of symmetry for the triangle? Why or why not?
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By the Triangle Symmetry Property, By the Triangle Symmetry Property, is the yellow segment an axis of is the yellow segment an axis of symmetry for the triangle? Why or symmetry for the triangle? Why or why not?why not?
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Rotational SymmetryRotational Symmetry
Defn: A figure has rotational Defn: A figure has rotational symmetry iff it has at least one symmetry iff it has at least one rotation image not counting rotation rotation image not counting rotation images of 0° or multiples of 360° images of 0° or multiples of 360° that coincide with the preimage.that coincide with the preimage.
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Rotational SymmetryRotational Symmetry
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Rotational SymmetryRotational Symmetry
All geometric figures have 0° (360°) All geometric figures have 0° (360°) rotational symmetry. If a figure rotational symmetry. If a figure has only 0° (360°) rotational has only 0° (360°) rotational symmetry, it is said to have trivial symmetry, it is said to have trivial rotational symmetry.rotational symmetry.
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Rotational SymmetryRotational Symmetry
n-fold rotational symmetry: A n-fold rotational symmetry: A figure that is said to have n-fold figure that is said to have n-fold rotational symmetry will rotate rotational symmetry will rotate onto itself n times if the figure is onto itself n times if the figure is rotated 360°.rotated 360°.
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Center of SymmetryCenter of Symmetry
The center of the rotation which The center of the rotation which yields the rotational symmetry is yields the rotational symmetry is also called the center of also called the center of symmetry.symmetry.
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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.
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AssignmentAssignment
Pages 143- 146,Pages 143- 146,
# 10 – 26 (evens), 28 – 32 (all), 34 – # 10 – 26 (evens), 28 – 32 (all), 34 – 40 (evens), 46 – 50 (all), 52, 5440 (evens), 46 – 50 (all), 52, 54