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Math Help - Points on a circle, calculating angle

  1. #1
    Senior Member Mukilab's Avatar
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    Points on a circle, calculating angle

    O is the centre of a circle
    A, B and C are points on this circle

    angle COA is 130 degrees
    Find the size of CBA

    I have no idea how to approach this question, answer is not necessary, merely how to approach it.
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  2. #2
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    Quote Originally Posted by Mukilab View Post
    O is the centre of a circle
    A, B and C are points on this circle

    angle COA is 130 degrees
    Find the size of CBA

    I have no idea how to approach this question, answer is not necessary, merely how to approach it.
    Hi Mukilab,

    it depends on whether or not B is on the short arc between A and C
    or on the longer arc.

    In both cases, you can use.....

    the angle at the centre of the circle is twice the angle at the circumference, for angles standing on the same arc.

    Hence angle CBA is found from angle COA.
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  3. #3
    Senior Member Mukilab's Avatar
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    CB and AB are the longer arcs, CO and AO are the shorter arcs.

    I'm confused by your explanation. What do you maen by the angle at the circumference?
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  4. #4
    A riddle wrapped in an enigma
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    Quote Originally Posted by Mukilab View Post
    O is the centre of a circle
    A, B and C are points on this circle

    angle COA is 130 degrees
    Find the size of CBA

    I have no idea how to approach this question, answer is not necessary, merely how to approach it.
    Which scenario are we talking about? The blue angle or the red angle?

    The angle at the circumference means an inscribed angle with its vertex on the circle's edge.
    Attached Thumbnails Attached Thumbnails Points on a circle, calculating angle-circlex.jpg  
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  5. #5
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    hi Mukilab,

    the arcs are the curved line segments of the circumference of the circle.

    The longer arc in the attachment goes from C around the circle to the left
    until reaching A.

    If B is on that side, then angle ABC is half of 130 degrees.

    However if B is on the shorter arc, then angle ABC is half of 230 degrees.

    The blue angles are the angles at the circumference,
    the black ones are the angles at the centre.
    Attached Thumbnails Attached Thumbnails Points on a circle, calculating angle-arcs.jpg  
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  6. #6
    Senior Member Mukilab's Avatar
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    Quote Originally Posted by Archie Meade View Post
    hi Mukilab,

    the arcs are the curved line segments of the circumference of the circle.

    The longer arc in the attachment goes from C around the circle to the left
    until reaching A.

    If B is on that side, then angle ABC is half of 130 degrees.

    However if B is on the shorter arc, then angle ABC is half of 230 degrees.

    The blue angles are the angles at the circumference,
    the black ones are the angles at the centre.
    Thank you, I'll accept your word, but I'm still confused as which arc we're talking about. In 'masters' diagram it's the red B
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  7. #7
    A riddle wrapped in an enigma
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    Quote Originally Posted by Mukilab View Post
    Thank you, I'll accept your word, but I'm still confused as which arc we're talking about. In 'masters' diagram it's the red B
    Well, if you're talking about the red B,
    then that angle is one-half the measure
    of its intercepted arc.

    What is the measre of its intercept arc, you ask?

    Well, it's the same as the measure of the central angle that also intercepted it.

    That central angle is angle COA which you originally stated as being 130 degrees.

    Therefore, angle CBA = 1/2(130) = 65 degrees.

    Remember this theorem: The measure of an inscribed angle in a circle is equal to one-half the measure of its intercepted arc.
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