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

PatternMan
Member
Registered: 2014-03-08
Posts: 199

Help?

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)

"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: 7,743

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

Children are not defined by school ...........The Fonz
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
Moderator
Registered: 2010-06-20
Posts: 7,743

Re: Help?

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

Children are not defined by school ...........The Fonz
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: 7,743

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

Bob

Children are not defined by school ...........The Fonz
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: 7,743

Re: Help?

Here it is.

Children are not defined by school ...........The Fonz
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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,156

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

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,434
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'
I'm not crazy, my mother had me tested.

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

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

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

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,018

Re: Help?

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
The knowledge of some things as a function of age is a delta function.

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

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

Re: Help?

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: Harlan's World
Registered: 2011-05-23
Posts: 16,018

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
The knowledge of some things as a function of age is a delta function.

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

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,156

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

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,434
Website

Re: Help?

ShivamS wrote:

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

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

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

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: Riemann Sphere
Registered: 2011-01-29
Posts: 24,434
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'
I'm not crazy, my mother had me tested.

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

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,156

Re: Help?

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

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

Re: Help?

Agnishom, of course not.

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

ElainaVW
Member
Registered: 2013-04-29
Posts: 571

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: Riemann Sphere
Registered: 2011-01-29
Posts: 24,434
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'
I'm not crazy, my mother had me tested.

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

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,156

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

ElainaVW
Member
Registered: 2013-04-29
Posts: 571

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: Riemann Sphere
Registered: 2011-01-29
Posts: 24,434
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'
I'm not crazy, my mother had me tested.

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

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,156

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

ElainaVW
Member
Registered: 2013-04-29
Posts: 571

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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,156

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.
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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