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Math Help - interesting circle problem

  1. #1
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    interesting circle problem

    I found this the other day online, and its fairly interesting:

    a circle is inscribed in a 36 degree sector of a circle with an area of 25(pi) cm squared. What is the circumfrance and area of the smaller circle?

    I'm having some trouble wrapping my head around it though. any help?
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  2. #2
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    Re: interesting circle problem

    I drew up a quick diagram. i think it should look about like this.interesting circle problem-photo-2-.jpg
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  3. #3
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    Re: interesting circle problem

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  4. #4
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    Re: interesting circle problem

    Hi,
    The attached solution computes the radius of the small circle and an expression for its area. I leave it to you to find a "reasonably" simple expression for the circumference.

    interesting circle problem-mhfgeometry23a.png
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  5. #5
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    Re: interesting circle problem

    Quote Originally Posted by johng View Post
    Hi,
    The attached solution computes the radius of the small circle and an expression for its area. I leave it to you to find a "reasonably" simple expression for the circumference.

    Click image for larger version. 

Name:	MHFgeometry23a.png 
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    radius of large circle = 5 since area is 25pi.By cosine rule I found that the sector chord = 3.1.Rest is easy
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  6. #6
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    Re: interesting circle problem

    bjhopper,
    The length of the chord for the large circle of radius 5 is {5(\sqrt5-1)\over2} or about 3.09. The rest may be "easy", but how does this chord length determine the radius of the inscribed small circle?
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  7. #7
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    Re: interesting circle problem

    Hi JohnG,
    Some construction is necessary.Draw a tangent at the point that the sector chords perpendicular bisector meets both circles.Extend the two radii to meet this tangent.This creats two isosceles triangles.Draw the chord of one 18 deg sector.Do you see how to proceed starting with 1/2 of 3.1 to find 1/2 the base of the larger isosceles triangle from which the radius of the small circle is calculated from 36 deg rt triangle. center of small circle lies on the intersection of two angle bisectors of the large triangle
    Last edited by bjhopper; April 29th 2013 at 05:26 PM. Reason: add word
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  8. #8
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    Re: interesting circle problem

    Hi bjhopper,
    All that you say is certainly true and the computed radius of the small inscribed circle via your suggestion is the same as in my original posting. However, the length of the chord plays no part in the computation. Here is said computation:

    interesting circle problem-mhfgeometry26.png
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  9. #9
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    Re: interesting circle problem

    Hi John G,I like your trig solution,I get the same result using ge0metry and simple trig.I do not know advanced trig
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