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Polygons At a website called PlanetMath, they define a polygon as a “ compact subset of the plane whose interior is contractible and whose boundary consists of finitely many line segments.” Recall our discussion concerning the role that definition can play. This is a general math definition. In the first edition of Discovering Geometry, a high school geometry text, the authors define a polygon as “ a closed geometric figure in a plane formed by connecting line segments endpoint to endpoint with each segment intersecting exactly two others. “ Activity 0: What is a polygon? Question 1: In your group, draw 4 examples of a polygon. Question 2: In your group, draw 4 examples of a planar shape that is not a polygon. Question 3: I will be passing a sheet around on which you need to draw your examples in the appropriate spot. I will then put this sheet on the document projector so that we can all see everyone’s examples. Question 4: What are the different ways that we can categorize the polygons? Write these down as we discuss them. Make sure to include an example of each category. Question 5: What are the important “parts” of a polygon? List them and label them in a sketch. Activity 1: Generating the regular polygons with a compass and protractor. There is an easy way to generate the regular polygons with a circle. Step 1: construct a circle with your compass Step 2: To make a regular polygon with n sides, plot n equally spaced points about the perimeter of your circle. How are you going to decide what equally spaced means? You need to be exact. Step 3: Connect these n points, there is your regular polygon. Question 1: Using circles, construct regular polygons with n=4, 8 sides. Hint: Can you fold the circle? Question 2: With the regular polygons you constructed measure the “central” angles (the vertex is the circle center and the rays go through a pair of adjacent vertices) with a protractor or with “common sense”. What is the connection between the number of sides a regular polygon has, 360 degrees, and the measure of each of central angle? Write an algebraic relationship. Question 3: Use the algebraic relationship you discovered in question 2 to fill in the following chart. n 3 4 5 6 7 8 9 10 11 12 20 Central angle measure Number of central angles Question 4: Using your chart in question 3, construct circles with your compass and the appropriate central angles with your protractors to form regular polygons with n=3, 5, 6 sides. Question 5: Why would we have to “estimate” with a polygon with n=7 sides? Activity 2: Interior and central angles Question 1: In activity 1, you looked at central angles. Find the interior angles of the polygons you constructed in activity 1. Explain how you did this. Question 2: What is the algebraic relationship between the interior angle and the “central” angle? Justify this relationship. Question 3: Use your work in question 1,2 to fill in the following chart: n 3 4 5 6 7 8 9 11 Interior angle measure Activity 3: Wrap-up Question 1: Do all polygons have central angles? Explain. Question 2: Do all polygons have interior angles? Explain. Question 3: Draw an example of each “type” of polygon- see the board for the listing of the different types. Label the parts of each. Question 4: Describe what a polygon is and is not. Make sure to look back at your work in activity 0. Question 5: Define polygon.