Portfolio work for the segments of a polygon.docx

1、Portfolio work for the segments of a polygonZhang Ruolan 11C The segments of a polygonIntroduction:Aim of this task is to find and prove a general relationship between the ratios of the sides and the ratio of the areas of two regular polygons. The relationship between these two regular polygons is o

2、ne of them is construct inside the other one. Way to do that is a line segment is drawn from each vertex to a point on the opposite side sp that the segment divides the side in the ratio of 1: n to create the inside regular polygons.This task starts from two equilateral triangle and then further to

3、non-equilateral triangles. Finally, generate relationship of any regular polygons.Two equilateral triangles:I drew one equilateral triangle ABC first, and a line segment is drawn from each vertex to a point on the opposite side so that the segment divides the side in the ratio 1:2 to create another

4、equilateral triangle DEF as showing right side.Figure 1Then, I repeat the same procedure as I did for Equilateral triangle ABD and DEF for two more different side ratios. For the first one, the ratio is 1:4 and for the second one, the ratio is 1:5, they show below as figure two and figure three. Fig

5、ure 2 Figure 3 In these three figures, they all have the same format of their side ratio 1: n. therefore, I put their measurement together to compare to try to find out the relationship between the ratios of the sides and the ratio of the areas of the triangles. By using GSP5, I found the side lengt

6、h of AB and DE, then I found the area of triangle ABC and DEF. Then, I try several different possible relationships between ratio of the sides and the ratio of the areas of the triangles. I tried to double side length or multiple them together first because the equation uses to calculate area of a t

7、riangle is A= (A=area of triangle). Finally, I found if I square sides length AB and DE first and then calculate the ratio of square sides length, which is, the ratio is the same as the ratio of the areas of the triangles, which is just as showing in the diagram below. Although there are slight diff

8、erences between them, this can prove my conjecture is right because there are some rounds off through the process and this caused slight difference. I can show my conjecture is right because the relationship I found between ratio of sides and the ratio of the areas of the triangle, which is= can be

9、proved by data. However, this cannot prove that this is a general relationship for all equilateral triangles, which is constructed in the same way (1: n). Therefore, I will use algebra to represent the relationship I found.As showing below, I use one of the sides ratios 1:2 to explain the relationsh

10、ip. The others can be proved by using the same method.Process:1. Construct height for both triangles, in, construct height from point B and BG is perpendicular to AC. In, construct height from point D and DH is perpendicular to EF.2. Because and are both equilateral triangles therefore, they have th

11、e same angles and they are similar to each other. Because they are similar to each other, these two triangles similarity equal to their height similarity equal to their sides similarity.The reason their height similarity equal to these two triangles similarity is because for AGB and EHD, they have t

12、wo pairs of same angles, angle BAG equal to angle DEH equal to sixty degree and angle AGB equal to angle EHD equal to ninety degree. They also have one pair of similar side AB and ED. Therefore, their height similarity equal to these two triangles similarity. 3. Because the equation uses to calculat

13、e area of a triangle is A=.For, A=ACBGFor, A=EFDHI use k to represent similarity of these two triangles.Therefore, =. In addition, their similarity equal to their height similarity equal to their sides length similarity equal to k. =4. Therefore, I prove my conjecture by this equation.Similarity of

14、these two triangles=similarity of their sides length=kSimilarity of these two triangles areas=This is the relationship between the ratios of the sides and the ratio of the areas of the triangles, =.Further investigation:According to information from question one; I do some further investigation, if

15、this relationship works for equilateral triangle, would it also work for non-equilateral triangle. Before doing the question, I assumed that this relationship should also work for non-equilateral triangle because this is a general relationship and there should not any exception of it.However, the re

16、lationship does not work for non-equilateral triangle when I drew it in GSP5. As showing in image one (please see below). By looking this example, I recognized that relationship between the ratios of the sides and the ratio of the areas of the triangles I came up through doing question one may has s

17、ome problems. Therefore, I did question one again to try to find out problems.By doing question one again, I found out the problem. In question two, the triangle that construct inside another triangle may not similar to outside one. For example, in figure one, I do not know weather similar to or not

18、 so I cannot use the relationship from question one because that relationship only work under narrow condition that is those two triangles have to similar to each other. The reason I did not recognize this in question one is that equilateral triangle always similar to each other, no matter how small

19、 or big they are. Therefore, the relationship I came out through question one is not a general relationship between the ratios of the sides and the ratio of the areas of the triangles because it only work for similar triangles.Therefore, I redo the question and found out the general relationship and

20、 my problem.Process:1. There is an equation to calculate the area of triangle, area=2. By using this equation, I get the same answer as question one, showing in image two.3. In addition, this works for non-equilateral triangles, using the triangle from image one as an example, showing in image three

21、.Therefore, this relationship, area= works for bother equilateral triangles and non-equilateral triangles.The reason I found= are equilateral triangles. Proving process showed below:Process:1. area=2. =3. Because triangle ABC and DEF both equilateral triangles, AB=BC, DE=EF and angle BAC=angle DEF=6

22、0 (degrees).4. Therefore, equation in process two equal to = and they satisfy the relationship, =.Therefore, for question two, the conjecture I made in question one is correct just it cannot eliminate the angle parts because they are not similar to each other Assumption:Therefore, I can make an assu

23、mption. No matter what kind of figures we have like triangles, squares, if they are similar to each other (like triangle ABC similar to triangle DEF, square ABCD similar to square FEGH), this relationship, = will fit them.To justify this assumption, I can repeat the same process I did in question on

24、e to construct and evaluate square.Therefore, according to all these data, I can tell the relationship between the ratios of the sides and the ratio of the areas of any regular polygons is the same as question one which is =.Limitation:The limitation of this is this relationship does not fit polygon

25、s, which are not similar to each other. Using triangles as an example, =If the triangles are not similar to each other, then I cannot cancel the angles as I did for similar triangles. Therefore, the restriction of this relationship is it does not work for polygons, which are not similar to each other.