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Math Help - Help needed with some circle (arcs, tangents, etc) relations

  1. #1
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    Help needed with some circle (arcs, tangents, etc) relations

    Sorry about this, I know people do not like solving my home work, but I would not have come here if I hadn't already spent an hour and a half searching on Google. Any way, I have a home work assignment and related test that involves the relations between circles, arcs, cords, tangent lines, secants, angles, etc. Since its too hard to describe as its a really complicated figure, I'll put a link to a Flickr page with a scanned version of the problem. If the photo is too small, hold ctrl and zoom in with the mouse wheel. Thanks guys.
    001 on Flickr - Photo Sharing!
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  2. #2
    MHF Contributor Amer's Avatar
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    you should know the following

    the radius which intersect with the tangent point is a perpendicular to the tangent line from that point

    I will find as much as I can
    so

    <ACO = 90
    <3 = 180 - <ACO = 90
    <ABO ( <5) = 90
    <COB = 360 - <ACO - <ABO - <CAB = 360 - 90-90-100 = 80
    <2 = 180 - <COB = 100
    <AOC = 180 - 70 = 110

    in triangle DOE OD = OE two radius so <4 = <OED
    so

    2<4 + 70 = 180 , <4 = 55

    <CED = 90 angle on the diameter

    <ECD + <4 + 90 = 180 , <ECD = 45

    in triangle BOD we have BO = DO why ?? so

    <OBD = < ODB = (180 - <2)/2 = 40

    <DCE = 1/2 <DOE center angle with circum angle at the same arc so

    <DCE = 35

    call the point of the intersection between CD and BF "x"
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  3. #3
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    Thanks, but I'm pretty sure O is not the center. I don't remember how, but because then AOB would be 100 degrees, if you look at it that would not work based on the properties I do know. I think the properties we're supposed to be using are diffirent. Our teacher had us write down 4 relations, but I lost them. The only one I remember is that the segments of externally secant lines through the same point cut by the perimeter are proportional.
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  4. #4
    A riddle wrapped in an enigma
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    Quote Originally Posted by chabos View Post
    Thanks, but I'm pretty sure O is not the center.
    Given: \odot O

    That pretty much tells you the center of the circle is O. And if you lost your notes, try reading the textbook.
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  5. #5
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    Yeah, you're right, sorry. I was thinking in parallelograms, so I thought all angles would have to be 90 of the kite. About the theorems, they're not really in the book. I finally found them, they are hidden within the problems themselves as proofs. Its stuff like the angle of the intersection of two tangent lines is 1/2 the major arc minus the minor.
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  6. #6
    A riddle wrapped in an enigma
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    Here are the answers I got.


    Angle 1: 80 degrees


    This angle is formed by two tangents.

    If two tangents intersect in the exterior of a circle, then the measure of the angle formed is equal to one-half the difference of the measures of the two intercepted arcs. 1/2(260-100)


    Angle 2: 80 degrees


    Angle AOB = 100 degrees because it is a central angle equal in measure to its intercepted arc.
    Angle 2 is supplementary to angle AOB because they make up a linear pair on diameter DC.

    Angle 3: 90 degrees

    A radius OC is perpendicular to a tangent (AC) at the point of tangency C.

    Angle 4 and Angle 5: 55 degrees

    Triangle DOE is isosceles with a vertex angle of 70 degrees.

    Angle 6: 90 degrees

    Same reason as Angle 3


    Angle 7: 65 degrees


    Angle 7 can be found by determining arcs BD and CF. Angle 7 is one-half the sum of these two arcs. 1/2(80 + 50)

    Arc BD = 80 because its the same measure as the central angle 2 which we found above.

    Arc DE is 70 - same as the central angle which was given.

    Arc CF can now be found by subtracting all the known arcs from 360. Arc CF is 50 degrees


    Angle 8: 35 degrees

    Angle 8 is one-half the measure of its intercepted arc DE.


    Angle 9: ?? I don't see an angle 9 in the picture!


    Arc BD: 80 degrees

    Same as the central angle that intercepted it.


    Arc BDE: 290 degrees

    Everything except the arc DE.
    Last edited by masters; April 29th 2010 at 07:28 AM.
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