Exploring Geometric Figures Using Cabri Geometry II
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1 Exploring Geometric Figures Using Cabri Geometry II Regular Polygons Developed by: Charles Bannister. Chambly County High School Linda Carre.. Chambly County High School Manon Charlebois Vaudreuil Catholic High School Rita Colonnello Vaudreuil Catholic High School Daphne Mullins Alexander Galt High School Cheryl Powell... Alexander Galt High School Resource People: Carole Bamford... MAPCO Carolyn Gould. MAPCO These materials were produced during the 98/99 school year under a PDIG grant of the Ministère de l'éducation de Québec involving teachers from the Riverside, Eastern Townships and L.B. Pearson School Boards
2 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 1 Regular Polygons using Cabri Terminal Objective 3.3: To solve problems involving polygons. Activity 1 Task: Explore relationships among Polygons and their Diagonals Construction: 1. From the Draw menu, select Show Axes. 2. From the Draw menu, select Define Grid, and click on one of the axes. 3. Select the Regular Polygon tool from the Lines menu and draw four different convex polygons; one in each quadrant. Quadrant 1: quadrilateral Quadrant 2: pentagon Quadrant 3: hexagon Quadrant 4: octagon
3 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 2 3. For each polygon, using the segment tool, draw all the diagonals, from a single vertex and determine how many triangles can be formed within each polygon. Polygon # of sides # of diagonals from one vertex # of triangles Quadrilateral Pentagon Hexagon Octagon 4. What do you notice about the number of sides and the number of triangles formed by the diagonals from one vertex in each polygon. Write your observations. What conclusion could be drawn about the number of triangles formed from a single vertex within a convex polygon with n sides?
4 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 3 Activity 2 Task: Explore relationships among interior angles of a Polygon Construction: Refer to your polygons in Activity #1 1. What is the sum of the measures of the interior angles of a triangle? 2. What is the sum of the measures of the interior angles of the 2 triangles using the quadrilateral in quadrant 1? 3. What is the sum of the measures of the interior angles of the 3 triangles using the pentagon in quadrant 2? 4. What are your observations? 5. Therefore, what is the sum of the measures of the interior angles of a: hexagon? octagon? decagon? 6. Complete the following table: Number of sides Number of triangles formed by drawing all possible diagonals from one vertex Sum of the measures of the interior angles x 180 =
5 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 4 7. Describe in words the relationship between the numbers of sides in a polygon and the sum of measures of its interior angles. What conclusion can be drawn about the sum of the measures of the interior angles of a polygon with n sides?
6 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 5 Activity 3 Task: Exploring the relationships among exterior angles of a Polygon An exterior angle of a polygon is formed when one of the sides is extended. Exterior angles lie outside the polygon. In this activity, you will discover the sum of the measures of the exterior angles of a polygon. Construction: 1. Using the Ray tool from the Lines menu, construct a hexagon with rays AB, BC, CD, DE, EF and FA. 2. On the rays, outside of the hexagon, construct points G on ray AB, H on ray BC, I on ray CD, J on ray DE, K on ray EF and L on ray FA. (The hexagon is probably not regular) 2. Measure the exterior angles: m LAB = m GBC = m HCD = m IDE = m JEF = m KFA =
7 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 6 3. Calculate the sum of these angle measures 4. Move parts of the hexagon to see if the sum changes (making sure your hexagon remains convex). Is the sum of these exterior angles the same for any hexagon? 5. Try similar constructions using rays to make triangles, quadrilaterals, pentagons, or other polygons with a set of exterior angles. What is the sum of the measures of one set of exterior angles in these polygons? What conclusion can be drawn about the sum of the exterior angles of a convex polygon?
8 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 7 Activity 4 Task: Exploring the Measure of the Central Angle in a Regular Polygon Construction: 1. Using the regular polygon tool, draw a hexagon anywhere on the screen. 2. Central angles are formed by joining each vertex of a regular polygon to the center. Using the segment tool, draw a radius from the center of your hexagon to each vertex. 3. What do you think is the sum of the central angles? 4. Measure each central angle and calculate the sum. What is your answer? 5. What is the relationship between the size of each central angle in a regular hexagon and the sum of the central angles?
9 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 8 6. Repeat the same steps to construct different regular polygons. In each case, what is the sum of the central angles? Record your answers in the following table: Polygon # of sides Sum of Central Angles Measure of each Central Angle Hexagon What conclusion can be drawn about the measure of the central angles of a regular polygon with n sides?
10 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 9 Activity 5 Task: Exploring the Perimeter of a Regular Polygon Construction: 1. Construct a regular pentagon. 2. What do you know about the lengths of the segments between any two consecutive vertices? 3. How is the perimeter of a regular polygon calculated, if the length of one side is known? 4. Calculate the perimeter of your pentagon. 5. Write the algebraic expression for the perimeter of each regular polygon in the table. Pentagon Hexagon Octagon Decagon Side Length Perimeter What conclusion can be drawn about the perimeter of a regular polygon with n sides of length c units?
11 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) 10 Activity 6 Task: Explore the Area of a Regular Polygon 1. Construct a regular octagon. 2. The line segment dropped from the center of a regular polygon and perpendicular to any one of the sides is called the apothem. It is usually denoted by the letter a. Find the midpoint of one of the sides of the regular octagon. Connect the center of the octagon to this midpoint. This is the apothem of your octagon. What is its measurement? 3. Measure the base of your octagon. What is its measurement?
12 Mathematics 216: General Objective 3 - Geometric Figures (Regular Polygons using Cabri) What is the formula used to calculate the area of a triangle? Calculate the area of this triangle. 5. Suggest a formula that could be used to calculate the area of your octagon (Remember that your octagon is made up of eight congruent isosceles triangles) Verify by multiplying the area of the triangle by 8 From the Measure menu, select Area and find the area of the octagon. Compare the two values that were just found. Are they the same? 6. How can the area of a regular polygon be calculated given its perimeter and apothem? Therefore, the formula to calculate the area of a regular polygon is:
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