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Math Help - circle geometry HELP!

  1. #1
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    circle geometry HELP!

    the question is: two circles meet at points X and Y. The points A and B of these circles are diametrically oppposite to the point A. Prove that the points A, B and Y are collinear.

    I don't even understand what diametrically opposite means....help!
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  2. #2
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    Diametrically-opposite means the two points are on opposite ends of a diameter. Meaning, the line connecting the two points is a diameter of the circle.

    A diameter passes through the center of the circle. A diameter bisects a circle, so the two sectors formed by a diameter have 180 degrees central angle each.
    Since the meaure of the arc subtended by a central angle is equal to the measure of the central angle, then a diameter bisects the circumference into two 180-degree arcs.

    The measure of an inscribed angle, meaning its vertex is on the circumference, is half of the measure of the circular arc subtended by the angle.
    So, the measure of an inscribed angle whose endpoints are diametrically opposite is (1/2)(180deg) = 90 degrees. --------------(i)

    In your Problem above, it should have been "points A and B are diametrically opposite to the point X", not the posted point A, so that points A,Y,B have a chance to be collinear.

    Draw the figure on paper. The two circles need not be equal , or the two circles may be different in sizes.
    Draw the points X,Y,A,B.
    Draw the line segment XY.

    In triangle XYB,
    angle XYB = 90 degrees

    In triangle XYA,
    angle XYA = 90 degrees

    Hence, angle AYB = 90 +90 = 180 degrees.
    So, AYB is a straight line.
    Therefore, A,Y,B are collinear. ----------------proven.
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  3. #3
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    Quote Originally Posted by lila_sine View Post
    ...

    I don't even understand what diametrically opposite means....help!
    Hello,

    I've attached a drawing to illustrate(?) ticbol's calculations.
    Attached Thumbnails Attached Thumbnails circle geometry HELP!-collinear_points.gif  
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  4. #4
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    THANKS ticbol for all your help and yes, I did a typo.
    THANKS earboth also for the diagram!

    However, ticboc with:

    Quote Originally Posted by ticbol View Post
    The measure of an inscribed angle, meaning its vertex is on the circumference, is half of the measure of the circular arc subtended by the angle.
    So, the measure of an inscribed angle whose endpoints are diametrically opposite is (1/2)(180deg) = 90 degrees. --------------(i)
    I really do not understand how you got 90 degrees.

    Well I looked at the problem again with the definition (thanks!) and I created a few more lines such as XY, C1Y and C2Y. Then from that since they were all isosceles triangles, I named angles A, B C and D. As the sum of a triangle is 180, A+B+C+D = 180. Maybe that was practically the same thing you were saying...except more mathematically but I just want to check if that proof is viable.

    Sorry for troubling you!
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  5. #5
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    Quote Originally Posted by lila_sine View Post
    THANKS ticbol for all your help and yes, I did a typo.
    THANKS earboth also for the diagram!

    However, ticboc with:



    I really do not understand how you got 90 degrees.
    ...
    Hello,

    ticbol was refering to the semicircle of Thales (that's the name I am used to for this theorem). I've marked the angles in question with 90.
    Attached Thumbnails Attached Thumbnails circle geometry HELP!-collinear_points2.gif  
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  6. #6
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    Quote Originally Posted by lila_sine View Post
    THANKS ticbol for all your help and yes, I did a typo.
    THANKS earboth also for the diagram!

    However, ticbol with:



    I really do not understand how you got 90 degrees.

    Well I looked at the problem again with the definition (thanks!) and I created a few more lines such as XY, C1Y and C2Y. Then from that since they were all isosceles triangles, I named angles A, B C and D. As the sum of a triangle is 180, A+B+C+D = 180. Maybe that was practically the same thing you were saying...except more mathematically but I just want to check if that proof is viable.

    Sorry for troubling you!
    No problem. I always welcome questions from the "asker" re my answers.

    Pardon me, but I cannot follow your angles A+B+C+D = 180 degrees.
    Also, triangles AC1Y and BC2Y being both isosceles has nothing do do with my solution.

    How I got 90 degrees?
    I thought I have explained it very well before I went for the solution.

    As you saw, I wrote this first:
    A diameter passes through the center of the circle. A diameter bisects a circle, so the two sectors formed by a diameter have 180 degrees central angle each.
    Since the meaure of the arc subtended by a central angle is equal to the measure of the central angle, then a diameter bisects the circumference into two 180-degree arcs.

    The measure of an inscribed angle, meaning its vertex is on the circumference, is half of the measure of the circular arc subtended by the angle.
    So, the measure of an inscribed angle whose endpoints are diametrically opposite is (1/2)(180deg) = 90 degrees. --------------(i)


    That's were I got my 90 degrees.

    earboth summarized the above by mentioning the Thales theorem.
    I have forgotten this theorem, although I know that any inscribed angle in a semicircle, whose rays terminate at the ends of the diameter, is a right angle or meausres 90 degrees.

    Go to Google and see for yourself what is Thales theorem. I just did.

    --------------------

    Thanks for the drawing and the Thales theorem, earboth.
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