# 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 $A$ to $N$. Between $AB$ there is a point $a$ such $Aa=aB$, between $BC$ there is a point $b$ such $Bb=\frac{1}{3}BC$, between $CD$ there is a point $c$ such $Cc=\frac{1}{4}CD$ and so on until complete the n vertex (builded).

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

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 $2$ to $7$ 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 $y=x$?

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
1 year, 6 months ago

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Sort by:

- 5 months, 1 week ago

Paola Ramirez posted this 1 year and 1 month ago.

- 5 months, 1 week ago

Yeah, I understand and I think she is not probably active on brilliant right now

- 5 months, 1 week ago

I agree with you. vasu paliwal, when did you start solving problems in brilliant? Are you new here?

- 5 months, 1 week ago