Fun with Diagonals. 1. Now draw a diagonal between your chosen vertex and its non-adjacent vertex. So there would be a diagonal between A and C.
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1 Name Date Fun with Diagonals In this activity, we will be exploring the different properties of polygons. We will be constructing polygons in Geometer s Sketchpad in order to discover these properties. To being this activity, open Geometer s Sketchpad on your desktop. Part A: Pre-Questions What is the sum of the interior angles of a triangle? Part B: Discovering the properties of polygons Quadrilaterals Step I: Draw a quadrilateral. 1. Using the segment tool on the tool bar to the left, create a quadrilateral and label the points counterclockwise A- D. See image below. Step II: Draw diagonals in the quadrilateral. 1. Now draw a diagonal between your chosen vertex and its non-adjacent vertex. So there would be a diagonal between A and C. Question 1: How many diagonals can you draw from this one vertex? Question 2: How many triangles are formed from the diagonal? Step III: Sum the interior angles 1. To find the sum of the interior angles, find the angle measures of each vertex and sum them together. a. To find the measure of vertex A, highlight point D, then point A and finally point B. Then go to the Measure drop down menu and select Angle. After doing this, you will see a pink highlighted box with the vertex in the middle and the measurement of the angle at that vertex in the upper left hand corner of the workspace. It will look like this.
2 b. Repeat part a for the angle measurement of vertices: B, C and D. 2. Now to find the sum of all the angle measurements, go to the Measure drop down menu and select calculate. A calculator will appear on your screen. Click on the first angle measure in the list of angle measures in the upper left corner of the workspace, then on the calculator choose the + sign and then select the next angle measure. Continue this process until you have the picture like the one to the right. Then press ok and you will see that the sum of the angles is highlighted in pink on your workspace. Drag your shape around to see if the sum changes. Question 4: What is the sum of the interior angles of a quadrilateral? Now draw a diagonal between B and D. To do this, use the segment tool again. See image below. Question 3: How many diagonals are there in the quadrilateral? Pentagons Step I: Draw a pentagon. Open a new blank document by going to the File drop down menu and selecting new blank document. 1. Using the segment tool on the tool bar to the left, create a quadrilateral and label the points counterclockwise A-E. See image below.
3 Step II: Draw the diagonals of the pentagon. 1. Now diagonals from A to the two non-adjacent vertices C and D. To do this, follow step 3 in the Quadrilateral section. Question 1: How many diagonals are there from the vertex A? Question 2: How many triangles are formed from the diagonals? Step III: Sum of the angles of the pentagon 1. To find the sum of the angles, follow the same steps for finding the sum of the angles in the quadrilateral section but do it for five vertices. Question 4: What is the sum of the angles of the pentagon? 2. Now draw diagonals from the other vertices of the pentagon to their non-adjacent vertices. Your figure should look similar to this picture: Question 3: How many total diagonals are there? Hexagons Step I: Draw a hexagon. Step II: Draw the diagonals of the hexagon. Step III: Find the sum of the angle measures. Answer the following questions: Question 1: How many diagonals are there from one vertex? Question 2: How many triangles are formed from the diagonals of the one vertex? Question 3: How many total diagonals are there in the hexagon? Question 4: What is the sum of the angles of the hexagon?
4 Pre-question: Can you predict the number of diagonals there are in a heptagon ( a polygon with seven sides)? Heptagon Step I: Draw a heptagon. Step II: Draw the diagonals of the heptagon. (It is getting harder to see the diagonals. Do your best to construct them) Step III: Find the sum of the angle measures. Answer the following questions: Question 1: How many diagonals are there from one vertex? Question 2: How many triangles are formed from the diagonals of the one vertex? Question 3: How many total diagonals are there in the heptagon? Question 4: What is the sum of the angles of the heptagon? Part C: Review Questions Fill in the table below with the information you found out about the diagonals of polygons. Polygon Number of sides (n) Number of diagonals from one vertex (v) Total Number of diagonals in the polygon (d) Triangle Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 1. What is the pattern you see between the number of sides of each polygon and the number of diagonals from one vertex of the polygon? 2. Can you determine the number of diagonals from one vertex given a decagon? How about for an nht-sided polygon?
5 3. Multiply this expression by the total number of sides n. Write the new expression. 4. Does this expression work to find the total number of diagonals? Why or why not? Check your expression by using the values from your table above. 5. If this does not work, what do you need to do to the expression to make it work? 6. Why do you think we need to do this? 7. Use your equation to find the number of diagonals, given a polygon with (you may use a calculator to compute) : a. 8 sides: b. 9 sides: c. 10 sides: 8. Fill in the table below with the information you collected during the activity Polygon Number of sides (n) Number of triangles made by diagonals Sum of interior angles Triangle Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 9. What is the pattern between the number of sides of each polygon and the number of triangles? 10. a. Can you determine the number of triangles given a decagon (a polygon with 10 sides)? b. How about an nth-sided polygon? 11. What is the pattern between the sum of the interior angles and the number of triangles? 12. a. What would the sum be for the interior angles of a decagon? b. How about an nth-sided polygon? 13. So to clarify, what is the general equation for finding the sum of the interior angles of a polygon?(i.e. If a
6 polygon as n sides, then the sum of its interior angles would be ) 14. Use your equation to find the sum of the interior angles of the polygons with the given sides below (you may use a calculator to compute) : a. 8 sides: b. 9 sides: c. 10 sides:
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Mathematics Concepts 2 Exam 1 Version 4 21 September 2018
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