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Math Help - Circles problem

  1. #1
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    Circles problem

    Three circles each with radius of 4, have centers A, E, and D, respectively. Two circles are tangent at E, and circles intersect at B (circle A and E) and C (Circle E and D). What is the perimeter of the quadrilateral ABCD (which seems to form a trapezoid).

    I know the answer is 20 but I am having trouble proving that BC is 4 as well.
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  2. #2
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    Re: Circles problem

    Quote Originally Posted by donnagirl View Post
    Three circles each with radius of 4, have centers A, E, and D, respectively. Two circles are tangent at E, and circles intersect at B (circle A and E) and C (Circle E and D). What is the perimeter of the quadrilateral ABCD (which seems to form a trapezoid).

    I know the answer is 20 but I am having trouble proving that BC is 4 as well.
    I have trouble picturing this, can you post a picture please?
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  3. #3
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    Re: Circles problem

    Quote Originally Posted by donnagirl View Post
    Three circles each with radius of 4, have centers A, E, and D, respectively. Two circles are tangent at E, and circles intersect at B (circle A and E) and C (Circle E and D). What is the perimeter of the quadrilateral ABCD (which seems to form a trapezoid).

    I know the answer is 20 but I am having trouble proving that BC is 4 as well.
    You are dealing with 3 equilateral triangles.

    BB'is the perpendicula bisector of AE; CC' is the perpendicular bisector of ED.

    Therefore BC = \frac12 AE + \frac12 ED

    Since |AE| = |ED| you'll get:

    BC = \frac12 AE + \frac12 ED = \frac12 AE + \frac12 AE = AE
    Attached Thumbnails Attached Thumbnails Circles problem-glschklg_trapez.png  
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