Investigating Relationships of Area and Perimeter in Similar Polygons


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1 Investigating Relationships of Area and Perimeter in Similar Polygons Lesson Summary: This lesson investigates the relationships between the area and perimeter of similar polygons using geometry software. Key words: Perimeter, area, similar, polygons Existing Knowledge Base: Prior to this lesson, students should understand similarity of polygons (the angles are congruent and the sides are proportional). Students are also expected to have some proficiency with a dynamic geometry software package. This lesson was designed specifically for Cabri II, but could be easily adapted for use with other software packages (such as Geometer s Sketchpad). NCTM Standards : NCTM Geometry Standard: Students should be able to analyze characteristics and properties of two and threedimensional geometric shapes and develop mathematical arguments about geometric relationships. Ohio Geometry and Spatial Reasoning Standard: Students identify, classify, compare and analyze characteristics, properties, and relationships of one, two, and threedimensional geometric figures and objects. Learning Objectives: Students determine that the ratio of the perimeters of similar polygons is equal to the ratio of any two corresponding sides, and that the ratio of the areas of similar polygons is equal to the square of the ratio of two corresponding sides. Materials: 1. Lab handout with detailed explanations of the activity 2. Cabri II 3. Calculator (if desired to make some calculations easier) Suggested Procedure: 1. As a motivator for the lesson, present the following problem to your students: You are purchasing index cards. A 3inch by 5inch pack of 100 cards costs $.98. How much should you expect a pack of inch by 6inch cards to cost? Have the students make an initial guess and explain that this problem will be revisited after the lab. 2. Students could be grouped in pairs for this procedure. 3. Step 4 in the student lab sheet is an extension of ideas and is designed for students who finish early and need extra work to challenge them. 4. After the class has determined a method for finding the ratios of perimeter and area of similar polygons, the original motivating question can be revisited, as well as several extensions. A worksheet with sample questions (including a revisit of the original motivating question) and an extension for after the class discussion has been included with this lesson. 5. Assessment for the lesson could be determined by checking the worksheet, and also by collecting and checking the lab sheets. One suggestion is that each student should have explanations in his or her own words
2 Investigating Relationships of Perimeter and Area in Similar Polygons Group Members names: File Names: Goal: The goal of this lesson is for each group of students to determine a relationship between the ratio of the sides of similar polygons and the ratios of perimeter and area. To make sure our calculations are not affected by rounding errors, we need to change a setting in Cabri II. Under the Options menu, select Preferences. Under the Display Precision and Units menu, set the Length option to 4. Step 1: Investigating Perimeter Relationships 1. Create a triangle. Label the vertices A, B, and C. (Use the Triangle tool) 2. Create a box containing the number.5. (Use the Numerical Edit tool) 3. Create a point way outside of your triangle. 4. Using the Dilation tool (under the sixth button of the tool bar), point at the triangle so that the cursor says, Dilate this triangle. Click the mouse once. Move the cursor to the point you drew outside your triangle until the cursor says with respect to this point and click the mouse again. Move the mouse toward the number.5 until the cursor says using this factor and click the mouse again. You should get a new triangle, smaller than the original. 5. The new triangle should not overlap the old triangle. If it does overlap, drag the point you dilated it with so that they no longer overlap. 6. Label the vertices of your new triangle D, E, and F, respectively. Is triangle DEF half as big as triangle ABC? Make a prediction. (We will verify the prediction later.) 7. Determine the perimeter and side lengths for each triangle. Label them on the screen for later use. Record your results below: (Use the Distance and Length tool) AB = BC = AC = DE = EF = DF = Perimeter of triangle ABC = Perimeter of triangle DEF =
3 If two triangles are similar, what do we know about the ratios of their corresponding sides? Now calculate the ratios: DE/AB = EF/BC = DF/AB = What relationship is there between the ratios of corresponding sides and the dilation factor we used? Why does this relationship exist? Can you predict what the ratio of the perimeters will be? Make a guess and write it here: Now, calculate the ratio of the perimeters. Perimeter of DEF/Perimeter of ABC = _ What relationship is there between the ratio of the corresponding sides and the ratios of the perimeters in similar triangles? Do you think this relationship will always work, or will it change if you change the triangle? Try dragging one of the vertices of the original triangle. Notice that this affects the similar triangle. Does the relationship you wrote down above still work? _ Why or why not? Save this file so you can use it later in the lab. Let s investigate a shape other than triangles. 1. Start a new file.
4 2. Create a square using (label it PQRS) and use the same method as above for creating a similar square (WXYZ, respectively), this time using a factor of.75 (Use Regular Polygon tool) Are all squares similar? 3. Determine the length of a side of each square and the perimeter of each square. Label these values. What is the ratio of the corresponding sides of the two squares? WX/PQ = What is the ratio of the perimeters of the squares? _ Does the relationship you determined for similar triangles also work for similar squares? What relationship is there between the ratio of the corresponding sides and the ratios of the perimeters in similar squares? Would you expect this to work for any similar quadrilaterals? What about other similar polygons? Why do you think this relationship exists? (Think about what you know about similarity of polygons and perimeters of polygons.) Step 2: Investigating Area Relationships Now we are ready to investigate the relationships involving ratios of sides and areas of similar polygons.
5 1. Go back to your first file (the one with the similar triangles). Determine the area of each triangle. (Use Area tool) 2. Create altitudes of your triangle and measure them. (Use Perpendicular Line tool) 3. You should have already determined the length of each side. Fill in the following information in the first row of the table below: DE AB DE/AB Altitude Of DEF Altitude of ABC Altitude DEF / Altitude ABC DEF ABC DEF / ABC At the beginning of the lab, we asked if DEF is half as big as ABC. Is it? How do we know this? 3. Now, drag a vertex of the larger triangle around so that the area of ABC is a different value. Notice that this changes the shape and area of DEF. Enter the information for this new triangle into the next row of the table. Do you notice any relationship between DE/AB, Altitude of DDEF/Altitude of DABC and DDEF/ DABC? You may need to play around with numbers a little. Try experimenting on a calculator to try to find a relationship that will work. When you come up with a relationship that works, write it below: 4. Go back to your file with the similar squares. Fill in the table below, first with your original values, and then after you drag a vertex: WX PQ WX/PQ WXYZ PQRS WXYZ/ PQRS
6 Do you notice any relationship between WX/PQ and WXYZ/ PQRS? Try checking if the relationship you wrote down in Step 3 works for similar squares. If that relationship does not work, try to come up with one that works for both triangles and squares. You may need to try relationships other than addition, subtraction, multiplication, and division. Consider using squares, square roots, cubes, cube roots, etc. When you come up with a relationship that works for both case (triangles AND squares), write it below: Step 3: Review and Conclusions Create a new file. Create a triangle and then create a similar triangle, using any dilation factor other than.5 or.75. Write down the dilation factor you used: You should be able to guess the ratio of corresponding sides. Remember to give your ratio in the form of new figure to old figure, since that is the order we have used all along. Make a guess and write it down: You should also be able to determine the ratio of perimeters of the triangles: Determine if your ratio is correct. Is it? Now, try to make a prediction for the ratio of the areas: Determine if your ratio is correct. Is it? Summarize your results below: Given the corresponding sides of similar polygons, the ratio of the perimeter is Given the corresponding sides of similar polygons, the ratio of the areas is
7 Step 4: Extension Will the relationships you discovered using triangles and squares work for other polygons? Experiment with both regular and irregular polygons. Create similar polygons and experiment with various factors of dilation. Fill in the chart below to keep track of your results. Number of Sides of Polygon Regular Or Irregular Ratio of Corresponding Sides Ratio of Perimeters Ratio of Areas Think about the following questions as you investigate: 1. Does the relationship you developed for squares and triangles still work for other regular polygons? What about irregular polygons? 2. Does it matter if my polygons are concave or convex? Write a summary of your findings in the space below:
8 Worksheet to Review Relationships of Area and Perimeter in Similar Polygons 1. Now that we have learned the relationships between corresponding sides of similar polygons and the ratios of their perimeter and area, we can revisit the original problem that motivated our discussion: You are purchasing index cards. A 3inch by 5inch pack of 100 cards costs $.98. How much should you expect a pack of inch by 6inch cards to cost? What about a pack of inch by 7inch cards? 2. Two similar heptagons have corresponding sides in a ratio of 3 inches: 7 inches. What is the ratio of their perimeters? What is the ratio of their areas? 3. Two similar pentagons have areas in the ratio of 25 cm 2 : 36 cm 2. What is the ratio of their sides? What is the ratio of their perimeters? 4. PQR has an altitude of 5.1 ft. LMN has an altitude of 6.2 ft. Given that the two triangles are similar, find the area of PQR given that the area of LMN is 9.3 ft 2 5. A regular decagon has an area of 90 square centimeters. A similar decagon has an area of 25 square centimeters. What is the ratio of the perimeters of the first decagon to the second? Extension: Hexagons ABCDEF and PQRSTU are similar. The scale factor of Hexagon ABCDEF to Hexagon PQRSTU is 14:3. The area of Hexagon ABCDEF is 24x. The area of Hexagon PQRSTU is x + 2. The perimeter of Hexagon ABCDEF is 40 + y and the perimeter of Hexagon PQRSTU is 2y11. a) Use the scale factor to find the ratio of the areas of the hexagons. b) Use the scale factor to find the ratio of the perimeters of the hexagons. c) What relationship is there between the ratio of the areas and the ratio of the perimeters? d) Solve for x and y. Explain your work. How did you know how to set up your equation? How could you check that your answer is correct?
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