Two-Dimensional Classifying Shapes
Polygons
Other Two Dimensional Figures
Conclusion
MATH 113 Section 8.2: Two-Dimensional Figures
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2007
Classifying Two-Dimensional Shapes
Polygons
Other Two Dimensional Figures
Conclusion
Outline
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Classifying Two-Dimensional Shapes Polygons Triangles Quadrilaterals Other Two Dimensional Figures Conclusion
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Classifying Two-Dimensional Shapes
Polygons
Other Two Dimensional Figures
Conclusion
Classifying Shapes
One of the important parts of geometry is classifying shapes and learning their properties. We begin our study of two dimensional gures with just such an exercise. Example How would you classify these shapes? List several dierent ways.
Classifying Two-Dimensional Shapes
Polygons
Other Two Dimensional Figures
Conclusion
Figures and Denitions
To help us nd standard classications for shapes, we start with a few denitions. Simple Closed Curves A simple closed curve is a curve which we can trace without going over a point more than once while beginning and ending at the same point. Example Which of the previous gures are simple closed curves? Questions What properties of a curve are being described by the terms: Simple Closed
Classifying Two-Dimensional Shapes
Polygons
Other Two Dimensional Figures
Conclusion
More Denitions
Lets examine several of these terms in more detail. Examining Each Term Curve - a straight or curvy set of connected points Closed - starting and ending at the same point Simple - does not cross itself Example Given these specically dened terms, draw each of the following.
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a simple closed curve a simple open curve a non-simple closed curve a non-simple open curve
Classifying Two-Dimensional Shapes
Polygons
Other Two Dimensional Figures
Conclusion
The Jordan Curve Theorem
Before we start talking about specic types of gures, we will look at one important general theorem in geometry. Jordan Curve Theorem Let c be a simple closed curve in the plane. Then the complement of the image of c consists of two distinct connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). History of the Theorem While 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 Veblen Rigorous formal proof of over 200,000 lines produced in 2005.
Classifying Two-Dimensional Shapes
Polygons
Other Two Dimensional Figures
Conclusion
Polygons
We start our detained exploration of two-dimensional gures with the polygon. The Polygon A polygon is a simple closed curve composed only of line segments. The line segments are called sides and the points where they meet are the vertices. Classication of Polygons Polygons can be classied by the number of sides. triangle - 3 sides quadrilateral - 4 sides pentagon - 5 sides hexagon - 6 sides heptagon - 7 sides octagon - 8 sides nonagon - 9 sides decagon - 10 sides
Classifying Two-Dimensional Shapes
Polygons
Other Two Dimensional Figures
Conclusion
Classifying Polygons
There are several general ways to classify polygons. Convex Polygons A convex polygon is one in which a line segment connecting any two points on the polygon lies completely inside the polygon. Concave Polygons In a concave polygon we can draw a line segment connecting two points on the polygon which lies in the exterior of the polygon. Regular Polygons A polygon in which all sides have the same length and all interior angles have the same measure is called regular. Diagonals A diagonal is a line segment which joins two non-adjacent vertices.
Classifying Two-Dimensional Shapes Triangles
Polygons
Other Two Dimensional Figures
Conclusion
The Triangle
The triangle with the fewest sides is of particular importance. Triangle A triangle is a polygon with exactly three sides. One of the important properties of triangles is that they are a stable, rigid structure. Classifying Triangles Triangles can be classied in several dierent ways. By sides - equilateral, isosceles, scalene. By angles - right, obtuse, acute Example Draw a Venn Diagram to show the interaction between right and isosceles triangles.
Classifying Two-Dimensional Shapes Triangles
Polygons
Other Two Dimensional Figures
Conclusion
The Median and Centroid
In the next few slides we will look at several types of line segments and the point at which they all intersect. The rst of these is discussed below. The Median The median of a triangle is the line segment that connects a vertex to the midpoint of the opposite side of the triangle. The Centroid As 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. Example Draw a triangle and nd its centroid.
Classifying Two-Dimensional Shapes Triangles
Polygons
Other Dimensional Two Figures
Conclusion
Perpendicular Bisectors and the Circumcenter
The next set of three line segments and their central point of intersection is made up of perpendicular bisectors. Perpendicular Bisector A perpendicular bisector is a line segment passing through the midpoint of a side which is perpendicular to that side. Circumcenter As there are three sides in a triangle, there are three perpendicular bisectors. These three line segments will always intersect at a single point called the circumcenter. The circumcenter is equidistant from the three vertices of the triangle. Example Draw a triangle and nd its circumcenter.
Classifying Two-Dimensional Shapes Triangles
Polygons
Other Two Dimensional Figures
Conclusion
Angle Bisectors and Incenters
Another center of a triangle can be located by nding the intersection of the angle bisectors. Angle Bisectors An angle bisector is a line segment which bisects (divides in two) an internal angle of a triangle. The Incenter As there are three angles in a triangle, there are three angle bisectors. These three line segments will always intersect at a single point called the incenter of the triangle. This is the point equidistant from all three sides. Example Draw a triangle and nd its incenter.
Classifying Two-Dimensional Shapes Triangles
Polygons
Other Two Dimensional Figures
Conclusion
The Altitudes and Orthocenter
The nal set of line segments and the center they dene is made up of altitudes. Altitude An altitude of a triangle is a line segment perpendicular to a side of the triangle and connected to the opposite vertex. The Orthocenter As there are three sides in a triangle, there are three altitudes. These three line segments will always intersect at a single point called the incenter of the triangle. Relationships The centroid, orthocenter, and circumcenter are all colinear. They are the same point only if the triangle is equilateral.
Classifying Two-Dimensional Shapes Triangles
Polygons
Other Two Dimensional Figures
Conclusion
Triangles and Congruence
Two polygons are congruent if all corresponding parts of the gure are congruent. That means corresponding sides have the same length, and corresponding angles have the same measure. Example Suppose two triangles, ABC and XYZ , are congruent. what does this mean about the line segments and angle measures of these triangles? Congruence in Triangles In your lab we will see several methods for showing that two triangles are congruent. They include: Side-Angle-Side Angle-Side-Angle Side-Side-Side
Classifying Two-Dimensional Shapes Quadrilaterals
Polygons
Other Two Dimensional Figures
Conclusion
Classifying Quadrilaterals
As with triangles, there are several ways to classify quadrilaterals, depending on angles and side lengths. Classifying Quadrilaterals The following are several types of quadrilaterals. trapezoid - at least one pair of parallel sides parallelogram - both pairs of opposi...