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#1 2014-04-23 06:11:55

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

Help?

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

nq4r.jpg

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


"School conditions you to reject your own judgement and experiences. The facts are in the textbook. Memorize and follow the rules. What they don't tell you is the people that discovered the facts and wrote the textbooks are people like you and me."

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#2 2014-04-23 06:24:54

bob bundy
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Registered: 2010-06-20
Posts: 6,426

Re: Help?

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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#3 2014-04-23 06:30:09

bob bundy
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Registered: 2010-06-20
Posts: 6,426

Re: Help?

Yes it does!  smile

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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#4 2014-04-23 06:41:23

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: Help?

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  dizzy

Bob


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

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#5 2014-04-23 06:42:59

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: Help?

Here it is.  smile

View Image: patternman.gif

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

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#6 2014-04-23 12:08:20

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Help?

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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#7 2014-04-23 12:58:22

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Help?

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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#8 2014-04-23 13:11:08

ShivamS
Member
Registered: 2011-02-07
Posts: 3,555

Re: Help?

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

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#9 2014-04-23 13:24:01

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Help?

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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#10 2014-04-23 13:25:10

ShivamS
Member
Registered: 2011-02-07
Posts: 3,555

Re: Help?

Man... I made the thread diverge again

I recall seeing it in the grade 9 NCERT book.

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#11 2014-04-23 13:25:56

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Help?

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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#12 2014-04-23 13:28:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Help?

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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#13 2014-04-23 13:32:34

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Help?

ShivamS wrote:

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

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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#14 2014-04-23 13:33:28

ShivamS
Member
Registered: 2011-02-07
Posts: 3,555

Re: Help?

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

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#15 2014-04-23 13:36:04

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Help?

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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#16 2014-04-23 13:36:40

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Help?

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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#17 2014-04-23 13:37:41

ShivamS
Member
Registered: 2011-02-07
Posts: 3,555

Re: Help?

Agnishom, of course not.

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#18 2014-04-23 13:39:05

ElainaVW
Member
Registered: 2013-04-29
Posts: 303

Re: Help?

What is bobbym talking about in post 12?

I know.

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#19 2014-04-23 13:39:53

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Help?

why are you saying that post 6 is experimental?


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#20 2014-04-23 13:42:42

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Help?

Hi ElainaVW;

Hi Agnishom;

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


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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#21 2014-04-23 13:52:41

ElainaVW
Member
Registered: 2013-04-29
Posts: 303

Re: Help?

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

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#22 2014-04-23 13:53:19

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Help?

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


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#23 2014-04-23 13:55:39

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Help?

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


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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#24 2014-04-23 13:59:58

ElainaVW
Member
Registered: 2013-04-29
Posts: 303

Re: Help?

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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#25 2014-04-23 14:00:49

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Help?

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


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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