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Math Help - Find Radius of Smaller Circle

  1. #1
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let O_1, \ O_2 be the centers of the circles. (The bigger circle has the center O_1), A\in(O_1), \ B\in(O_2)

    Then O_1A\perp AB, \ O_2B\perp AB.

    Let BC\parallel O_1O_2, \ C\in (O_1A)

    In the right triangle CAB apply Pitagora:

    BC^2=AC^2+AB^2\Rightarrow 400=(11-r)^2+19^2

    Now solve the quadratic. Remember that r<11
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  3. #3
    A riddle wrapped in an enigma
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    Hi magentarita,

    Ok, here's what I think.

    In my diagram I have drawn a line from the center of the larger gear to the tangent point of the smaller gear.

    Use the Pythagorean Theorem to find its length.

    c^2=11^2+19^2

    c=\sqrt{482}

    Find angle DBA using Arctan.

    \arctan \frac{19}{11}\approx 59.9314

    Angle BAC is also 59.9314 since alternate interior angles are congruent (BD and CA are parallel because we have two lines perpendicular to the same line)

    Now look at triangle ABC. We know c=\sqrt{482}, a = 20, and angle BAC = 59.9314.

    Use the Law of Cosines.

    a^2=b^2+c^2-2bc \cos A

    20^2=b^2 + (\sqrt{482})^2-2b(\sqrt{482}) \cos 59.9314

    This all boils down to


    b^2-22b+82=0

    Apply the quadratic formula to get your 2 results. One you have to throw away because it is bigger than the larger gear radius.

    Now you have your answer.
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  4. #4
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    Actually you are dealing with a right triangle. (See attachment)

    1. r = 11-x

    2. x^2+19^2=20^2

    Thus the correct answer is b)
    Attached Thumbnails Attached Thumbnails Find Radius of Smaller Circle-gemtang_an2krse.png  
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  5. #5
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    ow...

    I want to thank all of you for your reply.
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