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#1 2012-05-23 00:46:12

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Will someone tell me about this circle and chord relation?

In the adjoining figure,
Find angle BCD

Please give me the steps to do it

View Image: Circle.png

'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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#2 2012-05-23 01:25:48

anonimnystefy
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Re: Will someone tell me about this circle and chord relation?

First fin BAD from the triangle ABD,and get BCD from the fact that the quadrilayeral ABCD is inscribed in a circle,which means that BAD+BCD=180°.


“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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#3 2012-05-23 01:35:19

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Re: Will someone tell me about this circle and chord relation?

How do you know that BAD+BCD=180 degrees?
What is the property being used?


'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'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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#4 2012-05-23 04:02:19

bob bundy
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Registered: 2010-06-20
Posts: 7,020

Re: Will someone tell me about this circle and chord relation?

hi Agnishom

This is one of the five "angle properties of a circle"  **

see diagram

The angle at the centre is twice the angle at the circumference so

reflex BED = 2 x BAD

and

acute BED = 2 x BCD

As reflex BED + acute BED = 360  => BAD + BCD = 180.

This is true for both pairs of opposite angles in all cyclic quadrilaterals.

**Do you want to know the other four?

Bob

View Image: cyclic quad.gif

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

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#5 2012-05-23 22:54:54

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 21,569
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Re: Will someone tell me about this circle and chord relation?

Thanks Bob;
Now I understand
Will you kindly tell me the other properties?


'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'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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#6 2012-05-24 19:36:11

bob bundy
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Registered: 2010-06-20
Posts: 7,020

Re: Will someone tell me about this circle and chord relation?

hi Agnishom

(1)  In any circle the angle made by a chord AB at the centre is twice the angle made at the circumference.

In diagram 1, AOB = 2 x APB

proof:  Extend PO to C (exact position is not important)

Let x = APC and y = CPB

Triangle APO is isosceles (AO = PO = radius) => PAO  = x     =>  AOC = 2x

Similarly, COB = 2y.

Thus AOB = 2x + 2y = 2(x+y) = 2 x APB

(2)  Two angles on the same arc and made by the same chord will be equal.

As the angle AOB is fixed whilst AB is fixed, the point P may be moved to position Q with AQB = APB = half AOB

The proof of (1) breaks down if Q is moved so far round the circle that it moves the other side of A or the other side of B.  So Q = P only whilst Q is on the same arc as P.

If Q moves to the other side of the circle use property (4)

(3) If AB is a diameter then APB = 90.

obvious as 180 = 2 x 90

(4)  Opposite angles in a cyclic quadrilateral add up to 180.

Proof given in earlier post.

Note:  Given any three non collinear points, it is always possible to draw exactly one circle that goes through them.

If a fourth point is placed somewhere not on the circle a quadrilateral is made that is definitely not cyclic.

But, if the point is moved onto the circle, then property (4) applies.

Property (5) in next post.

Bob

View Image: angleprops.gif

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

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#7 2012-05-24 19:53:22

bob bundy
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Registered: 2010-06-20
Posts: 7,020

Re: Will someone tell me about this circle and chord relation?

Property (5)

The angle between a tangent and a chord is equal to the angle made by the chord on the opposite side of the circle.

In the diagram, TB is a tangent to the circle at B.  TBA = APB

proof;  AOB = 2x + 2y (see proof above for definition of x and y)

=> OBA = 90 - (x+y)

But OBT = 90  =>  TBA = x + y

Bob

View Image: angleprops2.gif

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

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#8 2012-05-24 20:04:23

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 21,569
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Re: Will someone tell me about this circle and chord relation?

Thanks


'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'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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#9 2012-05-24 23:23:13

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 15,922

Re: Will someone tell me about this circle and chord relation?

Isn't (3) just a special case of (1)?


“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 2012-05-24 23:50:41

bob bundy
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Registered: 2010-06-20
Posts: 7,020

Re: Will someone tell me about this circle and chord relation?

Yes, but it's useful to have it as a separate case.

B


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

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