Classifying Two-Dimensional ShapesPolygonsTrianglesQuadrilateralsOther Two Dimensional FiguresConclusionClassifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionMATH 113Section 8.2: Two-Dimensional FiguresProf. Jonathan DuncanWalla Walla UniversityWinter Quarter, 2008Classifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionOutline1Classifying Two-Dimensional Shapes2PolygonsTrianglesQuadrilaterals3Other Two Dimensional Figures4ConclusionClassifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionClassifying ShapesOne of the important parts of geometry is classifying shapes andlearning their properties. We begin our study of two dimensionalfigures with just such an exercise.ExampleHow would you classify these shapes? List several different ways.Classifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionFigures and DefinitionsTo help us find standard classifications for shapes, we start with afew definitions.Simple Closed CurvesA simple closed curve is a c urve which we can trace without goingover a point more than once while beginning and ending at thesame point.ExampleWhich of the previous figures are simple closed curves?QuestionsWhat properties of a curve are being described by the terms:SimpleClosedClassifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionMore DefinitionsLet’s examine several of these terms in more detail.Examining Each TermCurve - a straight or “curvy” set of connected pointsClosed - starting and ending at t he same pointSimple - does not cross itselfExampleGiven these specifically defined terms, draw each of the following.1a simple closed curve2a simple open curve3a non-simple closed curve4a non-simple open curveClassifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionThe Jordan Curve TheoremBefore we start talking about specific types of figures, we will lookat one important general theorem in geometry.Jordan Curve TheoremLet c be a simple closed curve in the plane. Then the complementof the image of c consists of two distinct connected components.One of these components is bounded (the interior) and the other isunbounded (the exterior).History of the TheoremWhile this may seem intuitively clear, it is not easy to show.First attempt by Bernard Bolzano in (1781-1848)Then by Camille Jordan (1838-1922)Finally proved in 1905 by Oswald VeblenRigorous formal proof of over 200,000 lines produced in 2005.Classifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionPolygonsWe start our detained exploration of two-dimensional figures withthe polygon.The PolygonA polygon is a simple closed curve composed only of line segments.The line segments are called sides and the points where they meetare the vertices.Classification of PolygonsPolygons can be classifie d by the number of sides.triangle - 3 sidesquadrilateral - 4 sidespentagon - 5 sideshexagon - 6 sidesheptagon - 7 sidesoctagon - 8 sidesnonagon - 9 sidesdecagon - 10 sidesClassifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionClassifying PolygonsThere are several general ways to classify polygons.Convex PolygonsA convex polygon is one in which a line segment connecting anytwo points on the polygon lies completely inside the polygon.Concave PolygonsIn a concave polygon we can draw a line segment connecting twopoints on the polygon which lies in the exterior of the polygon.Regular PolygonsA polygon in which all sides have the same length and all interiorangles have the same measure is c alled regular.DiagonalsA diagonal is a line segment which joins two non-adjacent vertices.Classifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionTrianglesThe TriangleThe triangle with the fewest sides is of particular importance.TriangleA triangle is a polygon with exactly three sides.One of the important properties of triangles is that they are a stable,rigid structure.Classifying TrianglesTriangles can be classified in several different ways.By sides - equilateral, isosceles, scalene.By angles - right, obtuse, acuteExampleDraw a Venn Diagram to show the interaction between right andisosceles triangles.Classifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionTrianglesThe Median and CentroidIn the next few slides we will look at several types of line se gme ntsand the point at which they all intersect. The first of these isdiscussed below.The MedianThe median of a triangle is the line segment that connects a vertexto the midpoint of the opposite side of the triangle.The CentroidAs there are three vertices in a triangle, there are three medians.These three line segments will always intersect at a single point,called the centroid (center of gravity) of the triangle.ExampleDraw a triangle and find its centroid.Classifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionTrianglesPerpendicular Bisectors and the CircumcenterThe next set of three line segm ents and their central point ofintersection is made up of perpendicular bisectors.Perpendicular BisectorA perpendicular bisector is a line segment passing through themidpoint of a side which is perpendicular to that side.CircumcenterAs there are three sides in a triangle, there are three perpendicularbisectors. These three line segments will always intersect at asingle point called the circumcenter. The circumcenter isequidistant from the three vertices of the triangle.ExampleDraw a triangle and find its circumcenter.Classifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionTrianglesAngle Bisectors and IncentersAnother center of a triangle can be lo cated by finding theintersection of the angle bisectors.Angle BisectorsAn angle bisector is a line segment which bisects (divides in two)an internal angle of a triangle.The IncenterAs there are three angles in a triangle, there are three anglebisectors. These three line segments will always intersect at asingle point called the incenter of the triangle. This is the pointequidistant from all three sides.ExampleDraw a triangle and find its incenter.Classifying Two-Dimensional Shapes Polygons Other Two Dimensional Figures ConclusionTrianglesThe Altitudes and OrthocenterThe final set of line segments and the center they define is madeup of altitudes.AltitudeAn altitude of a triangle is a line segment perpendicular to a sideof the