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Math Help - Circles and Quadrants Question

  1. #1
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    Circles and Quadrants Question

    Hi all. Befuddled for two nights because of this question. I seemed to have used all the tricks in the bag(e.g Pythagoras' Theorem, Cosine Rule, Area of Sector) but I still can't get past the 'showing' part of the question. And I have no idea how to start for the latter part of the question. So here goes the question:

    Circle ABC, centre P, touches the sides of quadrant ODE, centre O, at the points A,B and C and OD=6cm. Show the radius of the circle ABC is 2.49cm. Hence or otherwise, find the length of minor arc AC and the area of the shaded region.

    Circles and Quadrants Question-circle-question.png
    Hope to hear from you all soon. Thanks in advance.
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  2. #2
    Senior Member abhishekkgp's Avatar
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    Re: Circles and Quadrants Question

    Quote Originally Posted by AeroScizor View Post
    Hi all. Befuddled for two nights because of this question. I seemed to have used all the tricks in the bag(e.g Pythagoras' Theorem, Cosine Rule, Area of Sector) but I still can't get past the 'showing' part of the question. And I have no idea how to start for the latter part of the question. So here goes the question:

    Circle ABC, centre P, touches the sides of quadrant ODE, centre O, at the points A,B and C and OD=6cm. Show the radius of the circle ABC is 2.49cm. Hence or otherwise, find the length of minor arc AC and the area of the shaded region.

    Click image for larger version. 

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    Hope to hear from you all soon. Thanks in advance.
    prove that the line joining O and P passes through C. call the radius of the smaller circle r.then OP=??(in terms of r). note that PC =r and OC=OD=6. use OP+PC=OC.
    from the above you will be able to find the value of r and the rest would follow.
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  3. #3
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    Re: Circles and Quadrants Question

    Quote Originally Posted by abhishekkgp View Post
    prove that the line joining O and P passes through C. call the radius of the smaller circle r.then OP=??(in terms of r). note that PC =r and OC=OD=6. use OP+PC=OC.
    from the above you will be able to find the value of r and the rest would follow.
    Two questions:
    1) How do you prove that the line joining O and P will pass through C? Is there some kind of theorem?
    2) OP=r+?. I mean how do you find the length from O to the circumference of circle ABC. If we don't know it, we would have to take it to be another unknown, in which case we would have two unknowns, thus complicating matters.

    Hope to hear from you soon. Thanks for your quick reply.
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  4. #4
    Senior Member abhishekkgp's Avatar
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    Re: Circles and Quadrants Question

    Quote Originally Posted by AeroScizor View Post
    Two questions:
    1) How do you prove that the line joining O and P will pass through C? Is there some kind of theorem?
    note that the two circles have a common normal at the point C. the normal at a point of a circle passes through the centre of the circle. now do yo get it?
    2) OP=r+?. I mean how do you find the length from O to the circumference of circle ABC. If we don't know it, we would have to take it to be another unknown, in which case we would have two unknowns, thus complicating matters.
    OP=r*sqrt(2). this is because PA=r and OAPB is a square(why?)

    Hope to hear from you soon. Thanks for your quick reply.
    it doesn't add an extra unknown. if you have understood the above then you should get OC=OP+PC, which gives 6= r*sqrt(2) + r. which gives the value of r.
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  5. #5
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    Re: Circles and Quadrants Question

    Quote Originally Posted by abhishekkgp View Post
    it doesn't add an extra unknown. if you have understood the above then you should get OC=OP+PC, which gives 6= r*sqrt(2) + r. which gives the value of r.
    Thanks a million. I get the whole thing now. Since the sides of OAPB are the same(r), it is hence a square. Then I just use Pythagoras' Theorem to get OP: OA^2+AP^2=OP^2.
    Last edited by AeroScizor; June 14th 2011 at 02:17 AM.
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  6. #6
    Senior Member abhishekkgp's Avatar
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    Re: Circles and Quadrants Question

    Quote Originally Posted by AeroScizor View Post
    Thanks a million. I get the whole thing now. Since the sides of OAPB are the same(r), it is hence a square. Then I just use Pythagoras' Theorem to get OP: OA^2+AP^2=OP^2.
    spot on!!
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