# The segments of a polygon

INTERNATIONAL BACCALAUREATE

II. GIMNAZIJA MARIBOR

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.

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

== = 3 units

= =   1.13 units

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

Formula for area of triangle:

(u=unit)

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

(b) Repeat the procedure above for at least two other side ratios, 1:n.

I decided to draw triangle with ratio: 1:3

== = 3 units

= =  1.67 units

Third ratio: 1:4

== = 3 units

= =   1.96 units

(c) For:

n=2 we get ratio of areas 7

n=3 we get ratio of areas 3.2

n=4 we get ratio of areas 2.3

To see connection between this results the best way is to draw a graph on which n will be represented on ...