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Math Help - If the radius of the small circle is 8 cm, what is the circumference of the...

  1. #1
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    Question If the radius of the small circle is 8 cm, what is the circumference of the...

    I don't know how to do this question, could someone please explain?
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  2. #2
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    Hello, iamanoobatmath!

    I hope you can follow my labelling.
    (I couldn't make a diagram of this one!)

    The vertex of the angle is A.
    Draw a line from A through the centers of the circles.

    It first intersects the small circle at B,
    the center of the smaller circle is C,
    D is the point of tangency of the two circles,
    the center of the larger circle is E.

    From C, draw a radius down to the point of tangency, F.
    From E, draw a radius down to the point of tangency, G.

    We have: . BC = CD = CF = 8
    Let DE = EG = r

    In right triangle CFA,\;\angle CAF = 30^o\,\text{ and }\,CF = 8.
    . . Hence, the hypotenuse AC = 16.

    Since \Delta EGA \sim \Delta CFA, we have: . \frac{EG}{EA} \:=\:\frac{CF}{CA}

    . . That is: . \frac{r}{r+24} \:=\:\frac{8}{16} \quad\Rightarrow\quad r \:=\:24


    Therefore, the circumference of the larger circle is: . 48\pi cm.

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