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Given a polygon with known perpendicular lines from any point inside the polygon, make a statement about the height

 May 13, 2018
edited by Rollingblade  May 13, 2018
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Hi Rollingblade!

 

Melody made a similar answer here: https://web2.0calc.com/questions/weekly-cycle-challenge-2-help 

I was thinking about Melody's answer and was thinking about how to generalize it. Here is what I have:

 

Using the hexagon problem: 

 

The area of the hexagon is equal to the sum of the six triangles constructed from connecting the point inside the hexagon to the 6 vertices. 

 

\(S_{abcdef}=\frac12m(a+b+c+d+e+f)\)

 

The area can also be expressed as the sum of the areas of the 6 equilateral triangles. 

 

\(S_{abcdef}=\frac12\cdot6\cdot{m}\cdot{h}\)

 

Since these two expressions are equal, we get:

 

\(6h=a+b+c+d+e+f\)

 

This is true for any polygon. 

 

n-sided polygon:

 

\(n\cdot{h}=\overbrace{a+b+c+d+\dots+x+y+z}^\text{n-sides}\)

 

Where h is the height, and a - z are the known perpendicular lines from any point.

 May 13, 2018
edited by GYanggg  May 13, 2018

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