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Math Help - Trangle height formula?!

  1. #1
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    Exclamation Trangle height formula?!

    i need urgent help!!!

    i need a formula for this question...

    In this diagram, the area of a pentagon is 800 square centimeters.


    What is the area of the green part, in square centimeters? Please round up to the nearest whole number, and submit only the answer.






    i found the side length of the pentagon to be
    21.56cm


    please help me find the height of the green triangle?!
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  2. #2
    Senior Member DivideBy0's Avatar
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    Construct BE. Construct perpendicular bisector AC, extending it to D.

    Now, \angle BDE = \frac{180(5-2)}{5}=108^{\circ}, and since triangle BDE is isoscles (BD = DE), we can conclude that \angle EBD = \angle BED = \frac{1}{2}(180-108) = 36^{\circ}
    Now in triangle BCD,

    \cos{36^{\circ}}=\frac{BC}{21.5635622735}

    \Rightarrow BC = 21.5635622735\cos{36^{\circ}}=17.4452883385

    Now, we focus our attention on triangle ABC.

    \angle ABC = 108 - 36 = 72^{\circ}

    Now, \tan{72^{\circ}} = \frac{AC}{17.4452883385}

    \Rightarrow AC = 17.4452883385\tan{72^{\circ}}=53.6910767207

    Also, \cos{72^{\circ}}=\frac{17.4452883385}{AB}

    \Rightarrow AB = \frac{17.4452883385}{\cos{72^{\circ}}} = 56.4541389505

    Using similar triangles,

    \frac{AB}{AF} = \frac{AC}{AG}

    \frac{56.4541389505}{56.4541389505-21.5635622735}=\frac{53.6910767207}{AG}

    AG = 33.18

    You can continue from here.
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  3. #3
    Senior Member DivideBy0's Avatar
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    Haha, oh geeze, Soroban submitted a 3 line solution
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by DivideBy0 View Post
    Haha, oh geeze, Soroban submitted a 3 line solution
    Where?

    Ah! I see now.

    @himachuu: Please don't double post. See rule #1 here.

    -Dan
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