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May 12th, 2013, 11:32 PM   #1
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interesting!!

hello, here's the question
If A,D are ends of diameter of circle of radius r and B and C are on semicircle such that AB=BC=r/2 then find the ratio of CD to radius
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May 13th, 2013, 12:03 AM   #2
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Re: interesting!!

Quote:
Originally Posted by benten
hello, here's the question
If A,D are ends of diameter of circle of radius r and B and C are on semicircle such that AB=BC=r/2 then find the ratio :BD to radius
I'm getting



but i didn't use BC at all so i'm wondering if you have something else in mind.

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May 13th, 2013, 12:11 AM   #3
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Re: interesting!!

I am really sorry; it's CD to radius ratio.
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May 13th, 2013, 04:44 AM   #4
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Re: interesting!!

I got the answer to be 4*(2)^1/2, but I doubt 'cuz I don't know the answer.
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May 13th, 2013, 05:22 AM   #5
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Re: interesting!!

Let the radius of the circle be 1. Let the center of the circle be O. Then angle COD, in degrees, is



Using the Law of Cosines we calculate CD:



Hence the ratio of CD to the radius of the circle is 7:4.
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May 13th, 2013, 05:28 AM   #6
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Re: interesting!!

Thank you and how did you get the angle 180-4*sin^-1(1/4)?
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May 13th, 2013, 05:29 AM   #7
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Re: interesting!!

[color=#000000]I think that the result should be , I will post later.[/color]
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May 13th, 2013, 05:32 AM   #8
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Re: interesting!!

Okay, I got it!!
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May 13th, 2013, 05:38 AM   #9
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Re: interesting!!

but is my approach wrong....my approach is
if both chords belong to a regular polygon of n sides then n*(r/2) is greatest value less than 2*pi*r then n must be 12 and internal angle is 150 then answer is different
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May 14th, 2013, 02:04 AM   #10
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Re: interesting!!

[color=#000000]I will do the same as greg1313, here is a brief solution.

[attachment=0:33lnkz3n]greg.png[/attachment:33lnkz3n]

Using the law of cosines in the triangle



in the right triangle , . We can easily prove that the colored angles are equal, which proves that . So and finally .[/color]
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