11.3 Curves, Polygons and Symmetry

Transcription

1 11.3 Curves, Polygons and Symmetry

2 Polygons

3 Simple Definition A shape is simple if it doesn t cross itself, except maybe at the endpoints.

4 Closed Definition A shape is closed if the endpoints meet.

5 Polygon Definition A polygon is a simple closed curve where all sides are line segments. There are no arcs allowed.

6 Concave v. Convex Definition A polygon is convex if any segment connecting any two points in the interior of the curve lies entirely inside the curve. Definition A polygon is concave if any segment connecting two points of the interior of a curve passes outside of the curve

7 Classification of Polygons by Sides Number of Sides 3 Polygon Name

8 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle

9 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4

10 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5

11 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5 pentagon 6

12 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7

13 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8

14 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9

15 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10

16 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 11

17 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 11 hendecagon 12

18 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 11 hendecagon 12 dodecagon n

19 Classification of Polygons by Sides Number of Sides Polygon Name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 11 hendecagon 12 dodecagon n n-gon

20 Regular Polygons Definition A regular polygon is a polygon where all sides and interior angles are congruent. What is the name of a regular triangle?

21 Regular Polygons Definition A regular polygon is a polygon where all sides and interior angles are congruent. What is the name of a regular triangle? What is the name of a regular quadrilateral?

22 Regular Polygons Definition A regular polygon is a polygon where all sides and interior angles are congruent. What is the name of a regular triangle? What is the name of a regular quadrilateral?

23 Exterior Angle Sum What is the exterior angle sum for a square?

24 Exterior Angle Sum What is the exterior angle sum for a square? What is the exterior angle sum for a pentagon?

25 Exterior Angle Sum What is the exterior angle sum for a square? What is the exterior angle sum for a pentagon? Conjecture?

26 Exterior Angle Sum What is the exterior angle sum for a square? What is the exterior angle sum for a pentagon? Conjecture? Exterior Angle Sum The exterior angle sum for a convex polygon is 360

27 Triangle Classifications We can classify triangles in two ways - by the number of congruent sides and by the types of angles. What types of triangles are there?

28 Triangle Classifications We can classify triangles in two ways - by the number of congruent sides and by the types of angles. What types of triangles are there? Types of Triangles By Side By Angle scalene no congruent sides acute all angles acute isosceles at least two congruent sides right one right angle equilateral all sides congruent obtuse one obtuse angle

29 Triangle Classifications We can classify triangles in two ways - by the number of congruent sides and by the types of angles. What types of triangles are there? Types of Triangles By Side By Angle scalene no congruent sides acute all angles acute isosceles at least two congruent sides right one right angle equilateral all sides congruent obtuse one obtuse angle Important to note: your book correctly states that a triangle that is equilateral is also isosceles. Some books incorrectly say that these are disjoint descriptions.

30 Quadrilateral Classifications That we want to do here is list all properties we know about the different quadrilaterals. Included in these properties we want to point out are: if opposite sides are parallel if opposite sides are congruent if opposite angles are congruent if adjacent angles are congruent if adjacent sides are perpendicular if diagonals bisect each other if diagonals are perpendicular if diagonals are congruent if there is at least one pair of parallel sides if there is at least one pair of congruent sides

31 Parallelogram

32 Parallelogram opposite sides congruent opposite angles are congruent diagonals bisect each other opposite sides are parallel

33 Rectangle

34 Rectangle opposite sides congruent opposite angles are congruent diagonals bisect each other opposite sides are parallel

35 Rectangle opposite sides congruent opposite angles are congruent diagonals bisect each other opposite sides are parallel diagonals are congruent adjacent sides are perpendicular

36 Rhombus

37 Rhombus opposite sides congruent opposite angles are congruent diagonals bisect each other opposite sides are parallel

38 Rhombus opposite sides congruent opposite angles are congruent diagonals bisect each other opposite sides are parallel adjacent sides congruent

39 Square

40 Square opposite sides congruent opposite angles are congruent diagonals bisect each other opposite sides are parallel diagonals are congruent adjacent sides are perpendicular

41 Square opposite sides congruent opposite angles are congruent diagonals bisect each other opposite sides are parallel diagonals are congruent adjacent sides are perpendicular adjacent sides congruent

42 Kite

43 Kite two pairs of congruent adjacent sides diagonals are perpendicular one pair of congruent angles

44 Trapezoid

45 Trapezoid at least one pair of parallel sides

46 Isosceles Trapezoid

47 Isosceles Trapezoid at least one pair of parallel sides

48 Isosceles Trapezoid at least one pair of parallel sides at least one pair of congruent sides base angles congruent

49 Hierarchy for Triangles For triangles, here is the relationship between them based on sides. Triangle Scalene Triangle Isosceles Triangle Equilateral Triangle

50 Hierarchy for Quadrilaterals Can you come up with a similar diagram for quadrilaterals?

51 Hierarchy for Quadrilaterals Can you come up with a similar diagram for quadrilaterals? Quadrilateral Kite Trapezoid Parallelogram Isosceles Trapezoid Rhombus Rectangle Square

52 Symmetries 1 Line symmetry 2 Rotational symmetry 3 Point symmetry

53 Line Symmetry Line Symmetry A figure has line symmetry if we can draw a line that divides the figure in half. We can think of this as the line over which we can fold the figure to make it fold onto itself.

54 How Many Lines of Symmetry?

55 How Many Lines of Symmetry?

56 Rotational Symmetry Rotational (Turn) Symmetry A figure has rotational symmetry if we can rotate the figure some number of degrees less than 360 and get the same exact polygon back in the same orientation. We describe the property as α degrees of turn symmetry.

57 Rotational Symmetry?

58 Rotational Symmetry?

59 Point Symmetry Point Symmetry Point symmetry is the term that we use when a figure has 180 turn symmetry. Which of our figures have point symmetry?

60 Diagonals Diagonal A diagonal is a line segment that joins to nonadjacenet vertices.

61 Diagonals Diagonal A diagonal is a line segment that joins to nonadjacenet vertices. The question is, how many diagonals does a convex polygon have?

62 Diagonals Diagonal A diagonal is a line segment that joins to nonadjacenet vertices. The question is, how many diagonals does a convex polygon have? Number of Diagonals A convex polygon with n sides we have n(n 3) 2 diagonals.