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Math Help - Another Circle

  1. #1
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    Another Circle

    Circle with radius of 10 in
    Chord AB is 14 in
    Find measure of minor arc AB and then actual length of minor arc AB
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by stones44 View Post
    Circle with radius of 10 in
    Chord AB is 14 in
    Find measure of minor arc AB and then actual length of minor arc AB
    i'm a bit rusty on my circle theorems, so there may be an easier way to do this--but this is the first idea that came to me. see the diagram below

    we can use the law of cosines to find the angle x. once we have that, we can use the length of arc formula to find the desired arc
    Attached Thumbnails Attached Thumbnails Another Circle-circle.gif  
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    Hello, stones44!

    Here's another way to determine that central angle x = \angle AOB.

    In Jhevon's diagram, draw the apothem to chord AB.

    Then: . \sin\left(\frac{x}{2}\right) \:=\:\frac{7}{10}

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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Soroban View Post
    Hello, stones44!

    Here's another way to determine that central angle x = \angle AOB.

    In Jhevon's diagram, draw the apothem to chord AB.

    Then: . \sin\left(\frac{x}{2}\right) \:=\:\frac{7}{10}

    Haha, i didn't see that. that's nice. and a lot simpler than using the cosine rule! As always Soroban does it better
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    Quote Originally Posted by Jhevon View Post
    That's nice. and a lot simpler than using the cosine rule!
    While I agree Soroban's is standard and nice, I prefer the way you approached the problem.
    Look:  x = \arccos \left( {\frac{{200 - 14^2 }}{{200}}} \right).
    That is certainly calculator ready; and the actual arclength is 10x.
    Don't sell yourself short.
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    Hello, Jhevon!

    As always Soroban does it better.

    I certainly am not "better" or "smarter" than you . . . just lazier.
    . . I was not born with this gift; it took years of dedicated practice.

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    so actual arc length?

    how is it 10x?
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    Hello, stones44!

    so actual arc length?
    how is it 10x?

    The arc length formula is: . s \:=\:r\theta
    . . where r is the radius and \theta is the central angle in radians.

    Using Jhevon's approach: . \cos x \:=\:0.02\quad\Rightarrow\quad x \:\approx\:1.55 radians

    Therefore: . s \:=\:10(1.55) \:=\:15.5 inches.

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    is there another way because I haven't learned that way
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by stones44 View Post
    is there another way because I haven't learned that way
    which way are you refering to? Soroban's way or my way? or both?--God forbid
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    Quote Originally Posted by stones44 View Post
    is there another way because I haven't learned that way
    If you are referring to arc length, no.

    -Dan
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by stones44 View Post
    so actual arc length?

    how is it 10x?
    10 is the radius. and x is the angle that subtends the arc. it may have been better for me to call it \theta. so you see the arc length formula is s = r \theta, and that's why it is 10x, since r = 10 and \theta = x. there is no other method as far as this part is concerned. if it is everything up to this point that you are unclear on, please be specific as to what the problem is
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    Do you know the origin of the word radian? How large is one radian?
    One radian is the measure of the central angle in any circle that intercepts an arc of the circle equal in length to the radius of the circle. So in any circle, if we have a central angle of 2.5 radians the intercepted arc has length 2.5r. If the circle has radius 10 then any central angle of .35 radians intercepts an arc of length 3.5.
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    but if the measure of the arc is 88.8, meaning the central angle is 88.8, does that mean the length of the arc is 888? (with radius 10)
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by stones44 View Post
    but if the measure of the arc is 88.8, meaning the central angle is 88.8, does that mean the length of the arc is 888? (with radius 10)
    no. you have the angle given in degrees here. you would use a different formula for that. in a circle, there are 2 \pi radians, and no more. anymore than that and it means you are making more than 1 revolution around the circle.

    For degrees:

    The length of arc l is given by:

    l = \frac { \theta}{360} 2 \pi r

    where \theta is the angle that subtends the arc in degrees
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