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Math Help - [SOLVED] Modern Geometry: Tangents to Circles

  1. #1
    Member ilikedmath's Avatar
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    Exclamation [SOLVED] Modern Geometry: Tangents to Circles

    Given:



    Sides AB and BC of triangle ABC, extended, are tangent to a circle at points D and E, and side BC is tangent at F.

    If AB = 8, AC = 9, and CF = 2, find BF.

    My Work So Far:

    * I can see that BF = x and there are 2 x's, so if I connect point D to F, triangle BDF will be isosceles. But I don't really see how that will help me find the length I need.

    * Can I set up a proportion such that 8 : 2 = 9 : x?
    So x = 3 = BF?

    I'm stuck


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  2. #2
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    You could note: |AD|=|AE|~\&~|EC|=|CF|.
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  3. #3
    Member arpitagarwal82's Avatar
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    AD = AE
    => AB + BD = AC + CE
    => 8 + x = 9 + 2
    => x = 3
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  4. #4
    Member ilikedmath's Avatar
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    Quote Originally Posted by Plato View Post
    You could note: |AD|=|AE|~\&~|EC|=|CF|.
    Thank you! I did some research, and wow, had I known the fact that "2 tangents drawn to a circle from a point outside the circle are equal." then I would have gotten it right away. But I clearly don't remember our professor teaching that. Quite frustrating how they expect us to know something that wasn't taught: and we're paying for this kind of "education".

    Anyway, thank you very much!!!
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  5. #5
    Member ilikedmath's Avatar
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    Quote Originally Posted by arpitagarwal82 View Post
    AD = AE
    => AB + BD = AC + CE
    => 8 + x = 9 + 2
    => x = 3
    Awesome, thank you very much!
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  6. #6
    Member arpitagarwal82's Avatar
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    Quote Originally Posted by ilikedmath View Post
    Thank you! I did some research, and wow, had I known the fact that "2 tangents drawn to a circle from a point outside the circle are equal." then I would have gotten it right away. But I clearly don't remember our professor teaching that. Quite frustrating how they expect us to know something that wasn't taught: and we're paying for this kind of "education".

    Anyway, thank you very much!!!
    Class room education is good, but try to refer and go through the text books once on your own. This will give you a good grip on topic. maths is all about dedication and concentration

    Keep solving.
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  7. #7
    Member ilikedmath's Avatar
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    Quote Originally Posted by arpitagarwal82 View Post
    Class room education is good, but try to refer and go through the text books once on your own. This will give you a good grip on topic. maths is all about dedication and concentration

    Keep solving.
    thanks! yeah, it was my bad: i did look through the book and found the theorem there! sorry to my prof!
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