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Thread: [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 $\displaystyle AB$ and chord $\displaystyle AB$ is 30 m.
    Find the radius


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

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

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


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

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

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


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

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


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

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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 $\displaystyle |\angle OAB|=30^o$

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

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

    $\displaystyle 30m=r\sqrt{3}+\frac{4{\pi}r}{3}$
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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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