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HEXAGON Theory -- Tips to solve any HEXAGON Question

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HEXAGON Theory -- Tips to solve any HEXAGON Question [#permalink] New post 19 Aug 2013, 01:29
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Hi Guys,

We all come across Hexagon questions and get stuck.

Here are below few concepts to help you tackle any Hexagon question.

Note: Please see the figure side by side to understand this concept clearly.


First of all, What is a HEXAGON??
Attachment:
File comment: hexagon
hexagon.jpg
hexagon.jpg [ 50.09 KiB | Viewed 1213 times ]

It is six dimensional regular polygon.

Properties of HEXAGON:

No of sides - 6
No. of Diagonals - 9
Area of Hexagon- 3\sqrt{3}a^2/2 , where a is side of Hexagon.

Most people think that HEXAGON has only 3 diagonals( AD, BE and FC).
Well,This is absolutely incorrect.....

Lets prove it if you don't believe me.. :)

The no. of diagonals of any polygon can deducted from nC2- n
=> 6C2 -6
=> 15 -6
=> 9

As shown in the figure,

HEXAGON has 9 Diagonals( 6 long diagonals and 3 small diagonals)
Long Diagonals- AC,CE,EA,BF,FD and DB
Small Diagonals- AD,BE and FC

SECRET BEHIND HEXAGON:


Hexagon basically consists of 6 similar equilateral triangles.
In the figure, the six triangles are AGF, AGB ,BGC, GDC, GED and FGE.

Lets assume "a" is the side of HEXAGON as shown in the figure.

Now, AG = GD = a [ since equilateral triangles share sides of equal length "a"]

The length of the any small diagonal will be 2a, i.e for AD,BE and FC.

For Long Diagonals, lets try to calculate the value.

In triangle AFG,

Let AO is perpendicular bisector,
AO=a\sqrt{3}/2 [ the altitude in an equilateral triangle is a\sqrt{3}/2]

So FO=OG=a/2

Now, OC = a +a/2 => 3a/2

In right angle triangle AOC,
AC^2 = AO^2 + OC^2 [ Pythagorum theorem]
AC^2 = 3a^2/4 + 9a^2/4
AC^2 = 12a^2/4
AC^2= 3a^2

AC = a\sqrt{3}

For any HEXAGON will side "a",
The length of any long diagonal is a\sqrt{3} , i.e for AC,CE,EA,BF,FD and DB
The length of any small diagonal is 2a, i.e for AD,BE and FC.

Area of polygon= Sum of Area of six equilateral triangles
=> 6 * \sqrt{3}a^2/4
=> 3\sqrt{3}a^2/2

Thanks,
Jai

KUDOS if it HELPED..!!! :)
_________________

MODULUS Concept ---> inequalities-158054.html#p1257636
HEXAGON Theory ---> hexagon-theory-tips-to-solve-any-heaxgon-question-158189.html#p1258308

HEXAGON Theory -- Tips to solve any HEXAGON Question   [#permalink] 19 Aug 2013, 01:29
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