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Math Help - concentric circle problem

  1. #1
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    concentric circle problem

    I have no idea how to solve this. I love geometry, but I've found that I'm not very naturally good at it. please help!

    The circles in the figure I have drawn out are concentric. The chord AB is tangent to the inner circle and has a length of 12 cm. What is the Area of of the non-shaded region? (A of big triangle - A of small triangle)
    concentric circle problem-photo-4-.jpg
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  2. #2
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    Re: concentric circle problem

    Quote Originally Posted by aaronrpoole View Post
    The circles in the figure I have drawn out are concentric. The chord AB is tangent to the inner circle and has a length of 12 cm. What is the Area of of the non-shaded region? (A of big triangle - A of small triangle)
    Click image for larger version. 

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    As posed, I fear there is no unique answer to this question.
    Look at this webpage.

    I think that you need to know the radius of one of those two circles. Or some other term from that webpage.
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  3. #3
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    Re: concentric circle problem

    Hello, aaronrpoole!

    This is a classic problem . . . with a surprising punchline.


    The circles in the figure I have drawn are concentric.
    The chord AB is tangent to the inner circle and has a length of 12 cm.
    What is the area of of the non-shaded region? (Area of big circle - Area of small circle)
    Code:
                  * * *
              *           *
            *       C   6   *
         A *- - - - ♥ - - - -♥ B
                 *  |  *  o
          *     *  r|  o* R   *
          *     *   ♥   *     *
          *     *   O   *     *
                 *     *
           *        *        *
            *               *
              *           *
                  * * *

    O is the center of the circles.
    C is the midpoint of chord AB\!:\;CB = 6
    Let R = OB, the radius of the large circle.
    Let r = OC, the radius of the small circle.

    From right triangle BOC\!:\;r^2 + 6^2 \,=\,R^2 \quad\Rightarrow\quad R^2-r^2 \,=\,36 .[1]


    The area of the large circle is: \pi R^2
    The area of the small circle is: \pi r^2

    The area of the ring is: A \:=\:\pi R^2 - \pi r^2 \:=\:\pi(R^2-r^2)

    Substitute [1]: . A \:=\:\pi(36) \:=\:36\pi


    Surprise! .We didn't need to know the two radii.

    The small circle could be a golfball or the Earth.
    The area of the ring is constant!
    Thanks from johng
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  4. #4
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    Re: concentric circle problem

    While this subject can be very touchy for most people, my opinion is that there has to be a middle or common ground that we all can find. I do appreciate that youve added relevant and intelligent commentary here though. Thank you!
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