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# Geometry

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.A,B,C,D, and E are points on a circle of radius  2 in counterclockwise order. We know AB=BC=CD=DE=2.  Find the area of pentagon ABCDE.

Feb 4, 2022

#1
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Pentagon inscribed in a circle => 5 isosceles triangles with central angle 72o

Circle centre O with r = 2, angle subtended between A and B, B and C, C and D, D and E, E and A with centre O = 360o / 5 = 72o

Area of triangle ABO = (1/2)(2*2*sin 72o) = 1.902 sq units

[ABCDE] = There are 5 triangles of equal area = (1.902 sq units)5 = 9.51 sq units.

Feb 4, 2022
#2
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.A,B,C,D, and E are points on a circle of radius  2 in counterclockwise order. We know AB=BC=CD=DE=2.  Find the area of pentagon ABCDE.

Hello Guest!

$$360^o-4\cdot 60^o=120^o$$

$$ABCDE=4\cdot MAB+MEA\\ =4\cdot \frac{a\sqrt{3}}{2}+MEA\\ =4\cdot \frac{2\sqrt{3}}{2}+4\cdot sin(60^o)cos(60^o)\\ =4\cdot \frac{2\sqrt{3}}{2}+4\cdot \frac{1}{2} \sqrt{3}\ \cdot \ \frac{1}{2}\\ =5\cdot\sqrt{3}$$

$$ABCDE=8.660254$$

!

Feb 4, 2022
edited by asinus  Feb 4, 2022
#3
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$$ABCDE$$ forms an irregular pentagon, which can be split into $$5$$ triangles, each with an area of $$\sqrt3$$, so the answer is $$\color{brown}\boxed{5\sqrt3}$$

Feb 4, 2022
edited by BuilderBoi  Feb 4, 2022
edited by BuilderBoi  Feb 4, 2022
edited by BuilderBoi  Feb 4, 2022
edited by BuilderBoi  Feb 4, 2022
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ABCDE is NOT a regular pentagon so your logic is incorrect.

Melody  Feb 4, 2022
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Equal chords are subtended from equal angles. at the centre

We have 4 equilateral triangles.

together that make an angles at the centre of  4*60 = 240degrees

The angle at the centre remaining for the last tirangle is 120 degrees.

The Apothem for the 4 congruent trianlges is   sqrt3

So their total area is  4sqrt3

The last triangle is isosceles with angles 120, 30,30

The length of the base is 2sqrt3 and the height is 1 so ther area is    sqrt3

So the total area is  5sqrt3 units     (which agrees with asin's answer.)

Here is the pic

Feb 4, 2022
#5
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That's a cool drawing, how did you make it?

BuilderBoi  Feb 4, 2022
edited by BuilderBoi  Feb 4, 2022
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Hi BilderBoi,

Thanks!

I use a free program called GeoGebra.  It takes a while to get the hang of, just do simple things to start and your knowledge will build.

Here is a download page.  I didn't know that there were all these ones to choose from.  I just use Classic 6.

I don't think there used to be so many when i dowloaded mine.  ://

Anyway, I do almost all my diagrams from GeoGebra6  and I really enjoy using it.

I often prefer it even over Desmos.

If you give it a go I wll be more than happy to help when you get stuck :)

Melody  Feb 4, 2022
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Thanks! I've used it in class before, but I didn't know you do such stuff with it lol! (You misspelled my name btw.)

BuilderBoi  Feb 4, 2022
edited by BuilderBoi  Feb 4, 2022
#9
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oh if you have a free geogebra account you can save you pics online.

For posting all of my pics I used Gyazo.  It is like an onle version of 'Snip'

Gyazo has a free and a paid version.  They are the same except the paid version lets you store as many pics as you want but the free one stores only the most recent few.  I am not sure how many becasue I have the paid version.

I do not think Gyazo is the best online storage of pics, I expect there are many to chose from. This forum saves in Tinypics I think if you use the provided one.  But i used to find the link to tinypics was often broken, so Gyazo has been a better for me.

Melody  Feb 4, 2022
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Sorry BuilderBoi.  :)

Melody  Feb 4, 2022