Math Is Fun Forum

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)

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"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."

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

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

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Help?

Here it is. 

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.

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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.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

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

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.

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'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.

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Help?

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

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Help?

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

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“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.

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Help?

Man... I made the thread diverge again

I recall seeing it in the grade 9 NCERT book.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Help?

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

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“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.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Help?

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.

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'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.

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.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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?

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'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.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help?

Why don't we ask him?

---

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.

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Help?

Agnishom, of course not.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Help?

What is bobbym talking about in post 12?

I know.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Help?

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

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

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!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.