# Can anyone solve this math question?? I'm stuck!

### Can anyone solve this math question?? I'm stuck!

In trapezoid ABCD, AD||BC, E, F is the midpoint of AB and CD.$S_{\triangle ABD}:S_{\triangle BCD} = 3:7$. EF divides the trapezoid into top and bottom parts. Find the ratio of top part to bottom part.
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### Re: Can anyone solve this math question?? I'm stuck!

Let $BD\cap EF=L; \frac{S_{ADB }}{S_{BCD }}=\frac{3}{7}\Rightarrow S_{ABD }=3x; S_{BCD }=7x ; CF=FD ; AE=BE; AD|| BC\Rightarrow EF ||BC; EF||AD.$
$\Rightarrow \triangle LBE\approx \triangle DBA\Rightarrow \frac{S_{\triangle LBE}}{S_{\triangle DBA}}=(\frac{BE}{AB})^{2}\Rightarrow S_{\triangle LBE}=\frac{1}{4}S_{\triangle DBA} =\frac{3x}{4}$.
$\Rightarrow S_{DAEL }=S_{\triangle DAB }- S_{\triangle BEL}=3x-\frac{3x}{4}=\frac{9x}{4};$ similarly $\triangle DFL\approx \triangle DBC\Rightarrow S_{\triangle DLF}=\frac{7x}{4}$
$\Rightarrow S_{FLBC }=S_{\triangle DBC }- S_{\triangle DFL}=\frac{21x}{4}\Rightarrow S_{FCBE }=S_{FLCB }+S_{\triangle BEL }=\frac{21x}{4}+\frac{3x}{4}=\frac{a24x}{4}=6x ;$
$S_{DAEF }=S_{DAEL }+S_{\triangle DLF }= \frac{9x}{4}+\frac{7x}{4}=4x\Rightarrow \frac{S_{FCBE }}{S_{DAEF }}=\frac{6x}{4x}=\frac{3}{2}$.

nathi123

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