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**PatternMan****Member**- Registered: 2014-03-08
- Posts: 174

You are seeing 3 semi circles and one whole circle. what is the area of the shaded region in terms CD?

*Last edited by PatternMan (2014-04-23 06:12:38)*

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,395

Hi PatternMan

There's a theorem for circles (think it's called 'the rectangular property of a circle') as follows:

AD times DB = CD ^ 2

Does that help?

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
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Yes it does!

Write out areas in terms of AD and DB and subtract to get expression for green area. Lots of cancelling leaving you with an expression with AD.DB in it.

Proof of theorem coming up.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,395

See diagram below.

In any circle let A, B C and E be points anywhere on the circumference, and D be the point where AB crosses CE.

Consider triangles ACD and EBD.

angle ACD = angle DBE ( angles subtended by the same chord AE )

angle ADC = angle BDE ( vertically opposite angles )

Therefore ACD and EBD are similar triangles.

Your diagram is a special case where ADB is a diameter. => chord CE is bisected by ADB so CE = DE.

edit forgot to post the diagram

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
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Here it is.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,786

Hi;

We can also use Analytical geometry. We assign A = (0,0) and B = (x,0) and D = (x1,0) then

So obviously

The area of the largest semicircle is

The area of the rightmost semicircle is

The area of the leftmost semicircle is

The area we want is:

Massive cleanup ensues.

Now you notice that the part on the right is DC^2 so we can replace that.

So the area in question is DC^2 * pi/4.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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bob bundy wrote:

Hi PatternMan

There's a theorem for circles (think it's called 'the rectangular property of a circle') as follows:

AD times DB = CD ^ 2

Does that help?

Bob

By the way, We have used this property in lower grades to mark irrational numbers like sqrt(2) on the real number line.

I suspect this can be proven with Pythagoras too.

'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'

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,537

If I remember correctly, you learned that in grade 9... Is it already a lower grade?

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,532

We learned that in grade 6. Don't know about him.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,537

Man... I made the thread diverge again

I recall seeing it in the grade 9 NCERT book.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,532

Well, that does not mean they don't teach it earlier.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,786

Hmmm, but who knows one compelling reason why they should be teaching the method in post #6?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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ShivamS wrote:

If I remember correctly, you learned that in grade 9... Is it already a lower grade?

It is good enough. However, I am in Grade 11 now.

'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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,537

Oh really? I kept remembering 10. But I guess the school year just ended.

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Yes, it just ended. I am not at all saying that Grade 9 people are inferior.

What is bobbym talking about in post 12?

'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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,786

Why don't we ask him?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,537

Agnishom, of course not.

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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 291

What is bobbym talking about in post 12?

I know.

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why are you saying that post 6 is experimental?

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

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,786

Hi ElainaVW;

Hi Agnishom;

It is not what I think it is but what you think it is.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 291

It is experimental. He first used Geogebra to hammer out the details. Please post the Geogebra demonstration.

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That is rather cryptic, I do not get what you are talking about.

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

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,786

We are getting off track here. Patternman and I saw him a while ago is happy.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 291

why are you saying that post 6 is experimental?

http://www.mathisfunforum.com/viewtopic.php?id=19346

Most of that he got from me!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,786

That is true and I thank you. Sort of like the tail wagging the dog...

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**