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Math Help - The segment of a circle

  1. #1
    Junior Member Fnus's Avatar
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    The segment of a circle

    hello, any help would be appreciated (:

    'A sheep is tethered to a post which is 6 m from a long fence.
    The length of the rope is 9 m.
    Find the area which is available for the sheep to feed on'

    I know the formula to find a segment, but that'd require that I had the angle, and I dont, so I'm sorta at a loss...

    Thanks again for any help

    - fnus
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  2. #2
    Super Member
    earboth's Avatar
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    Quote Originally Posted by Fnus View Post
    hello, any help would be appreciated (:

    'A sheep is tethered to a post which is 6 m from a long fence.
    The length of the rope is 9 m.
    Find the area which is available for the sheep to feed on'

    ...
    Hello,

    1. draw a sketch (see attachment)

    2. to calcualte the angle \alpha use the properties of the right triangle:

    \cos(\alpha)=\frac{6}{9}=\frac{2}{3}~\Longrightarr  ow~\alpha \approx 48.2^\circ

    3. the area which is available for the sheep consists of a sector with the central angle 360^\circ - 2\cdot \alpha plus two right triangles.

    4. for confirmation only: I've got a total area of 226.6 mē
    Attached Thumbnails Attached Thumbnails The segment of a circle-sheepatrope.gif  
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  3. #3
    MHF Contributor
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    So you know how to find the area of a segment of a circle if you have the central angle of the segment.

    [Heck, yes, it is "segment". I called it "secant" in one of my replies here. ]

    So let me show you only how to find here that central angle then.

    Draw the figure on paper.
    Draw a circle of radius 9 meters. [Not actually 9 meters. Assume your radius is 9m.]
    Then draw a vertical secant or line that is 6m from the center of the circle.
    Then draw two radii, one radius each to the ends of the vertical chord (part of the vertical secant).
    Then draw a radius that passes through the midpoint of the chord. This particular radius is the perpendicular bisector of the said chord. It is also the bisector of the central angle of the minor sector, and so the minor segment (that segment on the other side of the fence) of the circle.

    [Don't get lost with the many terms or parts of the circle. I am beginning to be lost myself.]

    The central angle!
    There are two congruent right triangles formed. Any of the two has:
    ---hypotenuse (radius) = 9m
    ---vertical leg (half of the chord) = unknown
    ---horizontal leg = 6m
    ---central angle = say, theta.

    cos(theta) = 6/9
    So, theta = arccos(6/9) = 0.84106867 radians.

    Therefore, the central angle of the minor segment of the circle is 2(0.841069) = 1.682137 radians.

    Therefore also, the central angle of the major segment, the available area the cow can feed on, is 2pi - 1.682137 = 4.601048 radians.

    So now you have the two segments of the cirlce, with their corresponding central angles.

    You carry on?
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