Areas related with a linear equation

Some years ago, I posted two geometrical problems related with side-ratios in triangles and squares. This year, I was on my bed thinking when I remember this two problems and I try to generalize for regular polygons of n-sides. After some calculations in Geogebra, I found an interesting relation between the regular polygon (the original one) and the builded polygon which was a linear equation. The question from which the problem emerges is:

Let be a polygon of n sides (original) with vertex labeled from AA to NN. Between ABAB there is a point aa such Aa=aBAa=aB, between BCBC there is a point bb such Bb=13BCBb=\frac{1}{3}BC, between CDCD there is a point cc such Cc=14CDCc=\frac{1}{4}CD and so on until complete the n vertex (builded).

Then, polygons' areas are plotted (axis-xx for the area of builded polygons and axis-yy for the area of the original polygons) to values from n=1n=1 to n=17n=17

A linear equation is got it from the previous question.

For polygons of side one I got the areas in te table below which its graph

I did the same process for polygons of sides 22 to 77 and get the same linear equation. I'm intrigued about:

  • Why this data shows a linear equation?

  • How can I get a matemathical process to calculate the builded area?

  • Why the linear equations is so close from y=xy=x?

Those are my main questions, I hope someone get interested in this problem and together we could work together on this.

Note by Paola Ramírez
7 months, 3 weeks ago

No vote yet
1 vote

</code>...<code></code> ... <code>.">   Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in </span>...<span></span> ... <span> or </span>...<span></span> ... <span> to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


There are no comments in this discussion.


Problem Loading...

Note Loading...

Set Loading...