# Math IA- Type 1 The Segments of a Polygon

Math IA- Type 1

The Segments of a Polygon

Abhinav Jain

IB Higher Level Assignment: Internal Assessment

Type 1: Modeling

Mr. Murgatroyd

Date: 15/03/2009

Word Count:

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, creating another equilateral triangle DEF.

a)

i)

ii)

Measurements and drawing shown above has been made through the Geometer’s Sketchpad package.

Measure of one side of the ΔABC = 12cm

Measure of one side of the ΔDEF = 5cm

iii)

The areas have also been calculated using the Geometer’s Sketchpad package and are show in the diagram above.

For ΔABC = 62.5cm2

For ΔDEF = 8.8 cm2

In order to find the ratio, one needs to divide the area of ΔABC by the area of the ΔDEF which will give one the ratio of the area of the bigger triangle to the smaller triangle.

Therefore

62.5cm2 ÷ 8.8 cm2 = 7:1

The ratio between the areas of the equilateral ΔABC to ΔDEF when the segment divides the side in the ratio 1:2 is 7:1.

b) In order to repeat the procedure above for at least two other side ratios, 1: n

The two ratios chosen are 1:3 and 1:4 for no specific reason.

Ratio of Sides = 1:3

Again, the diagram above and values were obtained and created using Geometer’s Sketchpad Package.

For ΔABC = 62.0 cm2

For ΔDEF = 19.1 cm2

In order to find the ratio, one needs to divide the area of ΔABC by the area of the ΔDEF which will give one the ratio of the area of the bigger triangle to the smaller triangle.

Therefore

62.0 cm2/19.1cm2 = 13:4

The ratio between the areas of the equilateral ΔABC to ΔDEF when the segment divides the side in the ratio 1:3 is 13:4.

Ratio of Sides = 1:4

For ΔABC = 62.1 cm2

For ΔDEF = 26.6 cm2

In order to find the ratio, one needs to divide the area of ΔABC by the area of the ΔDEF which will give one the ratio of the area of the bigger triangle to the smaller triangle.

Therefore

62.1 cm2/26.6 cm2 = 7:3

The ratio between the areas of the equilateral ΔABC to ΔDEF when the segment divides the side in the ratio 1:4 is 7:3.

c) The following table compares the values of the ratios of the sides and the ratios of the areas of the triangles so that one can deduce a relationship by analyzing the results below.

If one looks at the values above, it is hard to be able to determine a relationship as the values of the ratio of areas are not very similar at all. However if one were to change the ratio of areas for the ratio of sides 1:4, from 7:3 to 21:9, then it would look like the following.

Now if one were to look at the values of ratios above, then one can see relationships both in the numerator and denominator of the ratios. One can see that the denominators are actually squares of integers such as 1, 2 and 3. The following table helps to analyze the situation better.

Looking over to the left of the table, one can see that as n increases, the value which is squared by which the denominator is found also increases in the same proportion which is 1. However, one also sees that the value which is squared to form the denominator is actually one less than n which illustrates the ratio produced by the segment that divides the ratio in its specific values. If one were to put this in mathematical terms, then it would be the following.

Value of denominator of ratio of areas of equilateral triangles = (n ...