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Math Help - Proofs! Externally Tangent Circles

  1. #1
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    Proofs! Externally Tangent Circles

    2 circles – one of radius 5, the other of radius 8 – intersect at exactly one point and the center of each circle lies outside the other circle. A line is externally tangent to both circles. Find the distance between the two points of tangency. Draw a figure and label it. Justify your answer completely with postulates and theorems. A two-column proof is suggested.









    THANKS SO MUCH FOR HELPING ME!!
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  2. #2
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    More generally:

    Consider two circles with radii r,R then the value of line externally tangent to both circles is given by 2\sqrt{r\cdot R}
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    thanks but can you be more specific and include proofs using postulates and theorems?
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    Quote Originally Posted by cbbplanet View Post
    thanks but can you be more specific and include proofs using postulates and theorems?
    Hello,

    I've attached a diagram of the situation. (The light grey circles are necessary to construct the tangent points)

    To calculate the distance between the tangent points you are dealing with a right triangle, which I've coloured grey.

    The length of the hypotenuse is (R + r)
    One leg (coloured violet) has the length (R-r)

    Now use Pythagorean theorem:

    d^2 + (R-r)^2 = (R+r)^2 Expand the brackets:

    d^2 + R^2 - 2rR + r^2 = R^2 + 2rR + r^2 . Subtract on both sides r² and R²:

    d^2 = 4rR~\implies~d = 2\sqrt{r \cdot R}
    Attached Thumbnails Attached Thumbnails Proofs! Externally Tangent Circles-abstd_beruehrpkte.gif  
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    I NEED AN EMERGENCY ASSISTANCE!!! the proofs are due tommorrow.


    i have figured everything out but can someone tell me how to prove the grey shade triangle is a right triangle? THANKS!!!
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by cbbplanet View Post
    I NEED AN EMERGENCY ASSISTANCE!!! the proofs are due tommorrow.


    i have figured everything out but can someone tell me how to prove the grey shade triangle is a right triangle? THANKS!!!
    One of its angle is the angles between a tangent and a radius to a circle and hence right.

    RonL
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