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The segments of a polygon

Extracts from this document...

Introduction

INTERNATIONAL BACCALAUREATE

II. GIMNAZIJA MARIBOR

image00.png

Portfolio mathematics HL

Assignment 1

The segments of a polygon

Author: Luka Dremelj

Candidate number:

Subject: Mathematics HL

Teacher: Barbara Pećanac

Date written: 25/5/2009

Introduction

First Mathematics HL Portfolio is about investigating the segments of a polygon, using graphical methods and analytical proofs. The task is to find conjecture between ratio of the sides and the ratio of the areas, first of triangles developing it to general conjecture of polygons.

1. In an equilateral triangle ABC, 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, crating another equilateral triangle DEF.

(a) What is the ratio of the areas of the two equilateral triangles? To answer this question,

        (i) create the above diagram  with your geometry package.

image01.png

(ii) measure the lengths of the sides of the two equilateral triangles

image37.png=image38.png= image39.png= 3 units

image40.png= image41.png= image42.pngimage43.png 1.13 units

(iii) find the areas of the two equilateral triangles and the ratio between them.

Formula for area of triangle: image93.png

image02.png                                                                        (u=unit)

image13.png

I got the same results for the areas, as before calculated with program.               (see the picture above)

image24.png

(b)

...read more.

Middle

=image37.png, image28.png=image40.png

y=image58.png=image59.png=image60.png

x=image61.png=image62.png=image63.png

p=image64.png=image65.png=image66.png

image67.png

To get the relationship between the ratios of the sides and the ratio of the areas I have to first get relationship between image57.png and image28.png. If I look on the first triangle above, I see that image68.png, so I have to calculate p, y and x to get image28.png.

Triangles image69.png and image70.png are similar (if two angles of one triangle are same to two angel in second triangle, they are similar). So I can get the relationship of sides of triangles.

image71.png

image72.png

image73.png

To calculate p I have to use cosine’s rule:

image74.png

Now I go again back to relations of sides in similar triangles to calculate x.

image75.png

image76.pngimage77.png

image78.png now I enter x that I get above into the equation for y and get:

image79.png

And now we have all components to calculate image28.png. We enter them in the equation(I have already wrote it down above) : image68.png

image80.png

image81.pngimage82.png  since image83.pngI get:

image84.pngimage85.png

So I prove the conjecture that I get from program in task c.

2. Does this conjecture hold for non-equilateral triangles? Explain.

image86.png

To easily understand this, I draw a non-equilateral triangle which sides are split in ratio 1:2.

...read more.

Conclusion

image28.png:

image29.png

image30.png

image31.png

I have all components of image28.png so I can calculate it by formula image16.png

image32.png

So I get the general formula for all regular polygons:

image34.pngimage26.png (s is number of sides)

To prove this formula I can put in data for my pentagon, which I get with program and compare ratio that I get with my formula and ratio that I get from results of program.

image35.png

and results from my formula

image36.png



Conclusion

Through four tasks I found the conjecture between the ratios of the sides and the ratio of the areas for all regular polygons. I found also some similarities in relationship between areas and sides in non-equilateral triangles, but as I didn’t find the analytical proof, I can’t claim that it holds for all of them.                        I find this task as very interesting and instructively because I learned a lot of new methods (graphical) how to solve problems using advanced computer programs (I also learned how to use these programs as well).  Furthermore I also refreshed my knowledge about geometry and came to some useful findings.

Software used:

GeoGebra(http://www.geogebra.org/cms/)

Graph 4.3 (http://www.padowan.dk/graph/)

MATLAB (http://www.mathworks.com/)

LUKA DREMELJ IB 2008/2010

...read more.

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