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Math Help - Similar Trianles,pythagoras

  1. #1
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    Similar Trianles,pythagoras

    In An Equilateral Triangle -'abc', 'd' Is A Point On 'bc' Such That Bd=1/3bc .prove That 9ad^2 =7 Ab^2
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  2. #2
    Super Member

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    Hello, Riny!

    I have a proof, but I don't know if it satisfies your needs . . .


    In equilateral triangle ABC, D is a point on BC such that BD = (1/3)BC.
    Prove that: .9·ADČ = 7·ABČ
    Code:
                    A
                    *
                   *:*
                  * : *
                 *  :  *
                *   :   *
               *    :    *
            6 *     :     * 6
             *      :      *
            *       :       *
           *        :        *
          *         :         *
         *          :E    3    *
        *-------*---*-----------*
        B - 2 - D - - - 4 - - - C
    Draw line segment AD.

    Let the side of the triangle be 6: .AB = BC = CA = 6
    . . Then: .BD = 2, .DC = 4

    Draw altitude AE to side BC.
    . . Then: .BE = EC = 3 and DE = 1


    Triangle AEB is a 30-60 right triangle.
    - - Hence: -AE = 3√3

    In right triangle AED: .ADČ .= .DEČ + AEČ .= .1Č + (3√3)Č .= .28


    And we have: .9·ADČ .= .9·28 .= .252
    . . . - - - and: .7·ABČ .= .7·6Č .= .252

    Therefore: .9·ADČ .= .7·ABČ

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