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        riddles >> medium >> Find The Radius
        
(Message started by: ThudanBlunder on Jan 18^th, 2009, 3:37pm)

Title: Find The Radius
Post by ThudanBlunder on Jan 18^th, 2009, 3:37pm

Given that the chords AC = CD = 3 and BD = 7, find the radius of the circle below.

Title: Re: Find The Radius
Post by Grimbal on Jan 18^th, 2009, 3:53pm

I find it easier to [hide]switch one 3 and the 7 line to make the 7 line horizontal.[/hide]
My result: [hide] ----> 9 <---- [/hide].

Title: Re: Find The Radius
Post by ThudanBlunder on Jan 18^th, 2009, 4:00pm

The resulting diameter? Anyway, a neat trick, Grimbal. I never thought of that. ::)

Title: Re: Find The Radius
Post by codpro880 on Jan 18^th, 2009, 8:05pm

[hide]4.5[/hide] ;D

Title: Re: Find The Radius
Post by Grimbal on Jan 19^th, 2009, 12:49am

on 01/18/09 at 16:00:12, ThudanBlunder wrote:

The resulting diameter? Anyway, a neat trick, Grimbal. I never thought of that. ::)

Err... yes, that would be the diameter. :-[

Title: Re: Find The Radius
Post by Barukh on Jan 19^th, 2009, 2:51am

on 01/18/09 at 16:00:12, ThudanBlunder wrote:

I never thought of that. ::)

It's really a very nice idea! Grimbal is ingenious, as always :D

Title: Re: Find The Radius
Post by Immanuel_Bonfils on Jan 19^th, 2009, 3:58pm

With Grimbal & Ptolemy it becomes cool.

Title: Re: Find The Radius
Post by chronodekar on Jun 24^th, 2009, 2:19am

If you switch the 3 and 7 lines, you get a trapezoid. Two of the lines are parallel, we know the lengths of the side-lines (pun not intended). But how does that give us the length of the bottom line? ???

I know that if you connect a point on a circle with the ends of a diameter, you get a right-angled triangle. Is there some similar law of geometry I'm missing here?

-chronodekar

Title: Re: Find The Radius
Post by Ronno on Jun 24^th, 2009, 5:59am

By Ptolemy's theorem, 3*3+7d=d^2-3^2

Title: Re: Find The Radius
Post by towr on Jun 24^th, 2009, 6:48am

For easy reference:
http://en.wikipedia.org/wiki/Ptolemy's_theorem (http://en.wikipedia.org/wiki/Ptolemy's_theorem)
http://mathworld.wolfram.com/PtolemysTheorem.html

Title: Re: Find The Radius
Post by chronodekar on Jun 24^th, 2009, 9:30am

I'll brush up on my geometry.

Thanks Ronno and towr.

-chronodekar

Title: Re: Find The Radius
Post by 1337b4k4 on Jun 24^th, 2009, 10:18am

Or if you don't remember much about Ptolemy like me,
[hide]extend BD to meet AC at point E. AOC, CDE and ABE are all similar, and the result quickly follows.[/hide]

Title: Re: Find The Radius
Post by ecoist on Jun 24^th, 2009, 5:42pm

I must be having a senior moment. I don't know Ptolemy and I don't know where 1337b4k4 goes after the similarities. Here's another solution.

[hide]Extend AC and BD to meet at E. Since AE and BC are orthogonal and angles ABC and CBE are equal, triangles ABC and BCE are congruent. Hence AE=6 and BE=d, where d is the diameter of the circle. Further, AD is orthogonal to BE, whence

(AD)^2=6^2 -(d-7)^2=d^2-7^2,

which simplifies to d^2-7d-18=0. Hence d=9.[/hide]

Title: Re: Find The Radius
Post by 1337b4k4 on Jun 26^th, 2009, 12:45pm

on 06/24/09 at 17:42:44, ecoist wrote:

I don't know where 1337b4k4 goes after the similarities.

[hide] http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/angle.gifAOD is twice http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/angle.gifABD, thus http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/angle.gifAOC is the same as http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/angle.gifABD, AOC and ABE are similar (two angles in common). Consequently, CE has length 3. http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/angle.gifACO, http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/angle.gifOCD, and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/angle.gifDCE add to 180, thus http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/angle.gifDCE is the same as http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/angle.gifAOC. As CDE is also isosceles, it is also similar to AOC and ABD. side lengths of similar triangles are proportional, thus r/3 = 3/(2r-7) so r = 4.5[/hide]

Title: Re: Find The Radius
Post by ecoist on Jun 26^th, 2009, 6:57pm

Danke sehr, 1337b4k4. Trivial details are often left out, but some of these details are not so trivial to me. Nice to know that there are those willing provide such details. I figured out your solution shortly after my post but I still struggled with your third similarity, triangle CED. Fortunately, that problem disappeared when I paid attention to the fact that the sum of angles CED and CDE equals angle ACD. Nice work, 1337b4k4, much better than my tedious solution.