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Could you show how I could demonstrate that a triangle with a fixed perimeter covers the maximum area when it is equilateral using Geogebra

May be we should Implement sliders

I downloaded Geogebra today itself, so I don't know much about it. Please explain it for a beginner

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Hi Agnishom;

Basically geogebra is for coming up with conjectures, not really for proving them in the rigorous sense.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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No problem but please show what I wanted in my first post of the thread

There is no need for a rigorous proof I know it

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

I can help you with the features of the program. I do not know if I can solve the problem with geogebra though. I am not an expert user.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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I think you are

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Thanks but I am not. I am working on the solution. It will take awhile.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Thank You for your efforts

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Hi Agnishom;

There is a nice demonstration using geogebra.

This is the finished product.

Want to see how it is done step by step with geogebra?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Yes, Please tell me

Also explain why the perimeter is fixed

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

We use one of the properties of the foci of an ellipse. That is what keeps the perimeter constant!

What version of geogebra have you downloaded?

Have you played a little bit with the proggie?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

To produce the demo do this:

1) In the input bar enter (2,2). Point A will be created.

2) In the input bar enter (10,2). Point B will be created. These will be the foci of the ellipse.

3) In the input bar enter (6,2+sqrt(48)). Point C will be created. This is to scale the ellipse correctly.

4) Use the ellipse tool and click A, B and C in that order and a nice ellipse will be created.

5) Hide C by right clicking it and unchecking show object.

6) Use the point on object tool to create D on the ellipse.

7) Move D around first with the mouse and then with your + and - keys. Notice it stays riveted to the ellipse.

8) Use the polygon tool and click A,B,D and back to A. poly1 will be created in the algebra pane and a number will be associated with it. This value is the area of the triangle.

9) Go into options-> rounding and click 5 decimal places.

10) In the algebra pane look under Segment and right click a. Then go into object properties and check show label. In the input box change it from name to value. Immediatey when you close the property box you will see the BD has a length for its label.

11) Repeat step 10 for b and d.

12) Use the move tool to position the length labels nicely around the triangle.

13) Enter in the input bar:

perimeter = a+b+d The variable "perimeter" will show in the algebra pane. It will have a value of 24.

14) Move D around and notice that though AD and BD change, the perimeter stays constant. This is a consequence of the definition of an ellipse.

15) Now comes the hard part. Move D around with your mouse watching poly1 closely. Try to get the largest value you can in poly1.

16) I got

poly1 = 27.7128

a = 7.99633

b = 8.00367

d = 8

This is highly suggestive that the largest area occurs at sides of length 8. Can we do a little better with the accuracy? Yes!

17) Right click D and go into properties -> Algebra and set increment to .001. Set in options rounding to 10 decimal places.

18) Hold down the shift key and press + or - until poly1 is the largest you can get it. On mine I get

poly1 = 27.7128129163

a = 7.9999258439

b = 8.0000741561

d = 8.

It does look like the equilateral triangle with sides of 8 is the best one. You should have something like figure 1. That ends the demonstration.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Will you please teach me "properties of the foci of an ellipse"?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Hi;

You will need a little background.

The foci are the two points that are used to create the ellipse.

Def: An ellipse is a locus (collection) of points such that: For any point P(x,y) on the ellipse, the sum of the distances to two given points F1 and F2 (the foci) is a constant.

http://www.valleyview.k12.oh.us/vvhs/de … guide.html

http://www.mathsisfun.com/geometry/ellipse.html

Like to see a video?

Also, please go through my geogebra tut post #11. It is especially for you and will answer and raise many questions. Geogebra is such a powerful tool for doing geometry. If you work with it you will have good results.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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I can understand it now

Thank You for your explanations

My bandwidth is very low to stream videos

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Hi Agnishom;

Anything else I can do? Did you get around to the maximum area tut? Need some help with it?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Thanks I don't need any more help about this.

I shall be using this image in one of my Maths project

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Hi Agnishom;

Let me know how you do.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Geogebra doesn't support Polar Equations, does it?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Yes, it does.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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How to use it then?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Hi Agnishom;

Do you have an example you would like shown?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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No, its better that YOU show an example

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Hi;

I am working on something simple that can be done on geogebra. Please hold on.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Okay

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,115

Hi;

I suppose the simplest is to show the polar grid and the creation of a single point A.

In polar coordinates geogebra specifies a point a little different than in cartesian.

1) Open geogebra and set the grid for the x and y axis to polar.

2) In the input bar type A = (3;1). The point A will be created in (r,θ) format. See the drawing below.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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