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Topic: Find The Radius (Read 2158 times) |
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ThudnBlunder
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Given that the chords AC = CD = 3 and BD = 7, find the radius of the circle below.
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Grimbal
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Re: Find The Radius
« Reply #1 on: Jan 18th, 2009, 3:53pm » |
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I find it easier to switch one 3 and the 7 line to make the 7 line horizontal.
My result: ----> 9 <---- .
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ThudnBlunder
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Re: Find The Radius
« Reply #2 on: Jan 18th, 2009, 4:00pm » |
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The resulting diameter? Anyway, a neat trick, Grimbal. I never thought of that. 
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| « Last Edit: Jan 18th, 2009, 4:13pm by ThudnBlunder » |
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codpro880
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Re: Find The Radius
« Reply #3 on: Jan 18th, 2009, 8:05pm » |
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4.5 
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Grimbal
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Re: Find The Radius
« Reply #4 on: Jan 19th, 2009, 12:49am » |
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on Jan 18th, 2009, 4:00pm, ThudanBlunder wrote:| The resulting diameter? Anyway, a neat trick, Grimbal. I never thought of that. |
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Err... yes, that would be the diameter. 
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Barukh
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Re: Find The Radius
« Reply #5 on: Jan 19th, 2009, 2:51am » |
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on Jan 18th, 2009, 4:00pm, ThudanBlunder wrote:| I never thought of that. |
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It's really a very nice idea! Grimbal is ingenious, as always 
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Immanuel_Bonfils
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Re: Find The Radius
« Reply #6 on: Jan 19th, 2009, 3:58pm » |
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With Grimbal & Ptolemy it becomes cool.
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chronodekar
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Re: Find The Radius
« Reply #7 on: Jun 24th, 2009, 2:19am » |
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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
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Ronno
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Re: Find The Radius
« Reply #8 on: Jun 24th, 2009, 5:59am » |
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By Ptolemy's theorem, 3*3+7d=d^2-3^2
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chronodekar
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Re: Find The Radius
« Reply #10 on: Jun 24th, 2009, 9:30am » |
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I'll brush up on my geometry.
Thanks Ronno and towr.
-chronodekar
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teekyman
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Re: Find The Radius
« Reply #11 on: Jun 24th, 2009, 10:18am » |
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Or if you don't remember much about Ptolemy like me,
extend BD to meet AC at point E. AOC, CDE and ABE are all similar, and the result quickly follows.
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ecoist
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Re: Find The Radius
« Reply #12 on: Jun 24th, 2009, 5:42pm » |
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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.
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.
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teekyman
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Re: Find The Radius
« Reply #13 on: Jun 26th, 2009, 12:45pm » |
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on Jun 24th, 2009, 5:42pm, ecoist wrote:| I don't know where 1337b4k4 goes after the similarities. |
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 AOD is twice ABD, thus AOC is the same as ABD, AOC and ABE are similar (two angles in common). Consequently, CE has length 3. ACO, OCD, and DCE add to 180, thus DCE is the same as AOC. 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
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| « Last Edit: Jun 26th, 2009, 12:45pm by teekyman » |
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ecoist
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Re: Find The Radius
« Reply #14 on: Jun 26th, 2009, 6:57pm » |
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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.
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