May 12th, 2013, 11:32 PM  #1 
Newbie Joined: May 2013 Posts: 26 Thanks: 0  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 
May 13th, 2013, 12:03 AM  #2  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: interesting!! Quote:
but i didn't use BC at all so i'm wondering if you have something else in mind.  
May 13th, 2013, 12:11 AM  #3 
Newbie Joined: May 2013 Posts: 26 Thanks: 0  Re: interesting!!
I am really sorry; it's CD to radius ratio. 
May 13th, 2013, 04:44 AM  #4 
Newbie Joined: May 2013 Posts: 26 Thanks: 0  Re: interesting!!
I got the answer to be 4*(2)^1/2, but I doubt 'cuz I don't know the answer.

May 13th, 2013, 05:22 AM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond  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. 
May 13th, 2013, 05:28 AM  #6 
Newbie Joined: May 2013 Posts: 26 Thanks: 0  Re: interesting!!
Thank you and how did you get the angle 1804*sin^1(1/4)?

May 13th, 2013, 05:29 AM  #7 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: interesting!! [color=#000000]I think that the result should be , I will post later.[/color] 
May 13th, 2013, 05:32 AM  #8 
Newbie Joined: May 2013 Posts: 26 Thanks: 0  Re: interesting!!
Okay, I got it!!

May 13th, 2013, 05:38 AM  #9 
Newbie Joined: May 2013 Posts: 26 Thanks: 0  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 
May 14th, 2013, 02:04 AM  #10 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  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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