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Thread: area of qudirilateral

  1. #1
    Apr 2009

    area of qudirilateral

    In triangle ABC, E is a point on AC and F is a point on AB. The lines BE and CF intersect at D. If the areas of triangles BDF, BCD nad CDE are 3,7 and 7 respectively, what is the area of the quadrilateral AEDF
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  2. #2
    MHF Contributor red_dog's Avatar
    Jun 2007
    Medgidia, Romania
    $\displaystyle area(BDC)=area(CDE)\Rightarrow BD=DE\Rightarrow area(ABD)=area(ADE)$

    Let $\displaystyle area(ADE)=x, \ area(ADF)=y$. Then, $\displaystyle x=y+3$

    $\displaystyle \frac{area(BDF)}{area(BDC)}=\frac{FD}{DC}=\frac{3} {7}=\frac{area(AFD)}{area(ADC)}$

    Then $\displaystyle \frac{y}{x+7}=\frac{3}{7}\Rightarrow 7y=3x+21$

    Solving the system $\displaystyle \left\{\begin{array}{ll}x=y+3\\7y=3x+21\end{array} \right.$ we get $\displaystyle x=\frac{21}{2}, \ y=\frac{15}{2}$

    $\displaystyle area(AEDF)=x+y=18$
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