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Math Help - [Arcs, sectors and segments] Finding the arc

  1. #1
    Junior Member Cthul's Avatar
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    [Arcs, sectors and segments] Finding the arc

    "A circle is modified so that it will measure a total of 30m around the major arc AB and the chord AB"
    Find the radius

    I don't know where to start.
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  2. #2
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    Hello, Cthul

    The perimeter of the shape formed by major arc AB and chord AB is 30 m.
    Find the radius


    Code:
                  * * *
              *           *
            *               *
           *                 *
    
          *         O         *
          *         ♥         *
          *    r  *120*  r    *
                *       *
           *  *           *  * 
          A ♥ - - - - - - - ♥ B
              *           *
                  * * *

    The circumference of the circle is: 2\pi r.

    The major arc has length: \frac{2}{3}\times 2\pi r \:=\:\frac{4\pi}{3}r


    Using the Law of Cosines on \Delta AOB\!:

    . . AB^2 \:=\:r^2 + r^2 - 2(r)(r)\cos120^o \:=\:3r^2

    Hence: . AB \:=\:r\sqrt{3}


    The perimeter is: . \frac{4\pi}{3}r + r\sqrt{3} \:=\:30

    . . . . . . . . . . . . \left(\frac{4\pi}{3} + \sqrt{3}\right)r \:=\:30


    Therefore: . r \;=\;\dfrac{30}{\frac{4\pi}{3} + \sqrt{3}} \;=\;\dfrac{90}{4\pi + 3\sqrt{3}}\text{ meters}

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  3. #3
    MHF Contributor
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    Quote Originally Posted by Cthul View Post
    "A circle is modified so that it will measure a total of 30m around the major arc AB and the chord AB"
    Find the radius

    I don't know where to start.
    Alternatively, split the triangle OAB into two identical right-angled triangles,
    where O is the circle centre.

    Then |\angle OAB|=30^o

    |AB|=2rcos30^o=r\sqrt{3}

    The length of major arc AB is \frac{240}{360}{2\pi}r

    30m=r\sqrt{3}+\frac{4{\pi}r}{3}
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  4. #4
    Junior Member Cthul's Avatar
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    Thanks to both, very helpful steps.
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