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

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

Postby Guest » Thu May 10, 2018 6:24 am

In trapezoid ABCD, AD||BC, E, F is the midpoint of AB and CD.[tex]S_{\triangle ABD}:S_{\triangle BCD} = 3:7[/tex]. 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!

Postby nathi123 » Fri May 11, 2018 12:38 pm

Let [tex]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.[/tex]
[tex]\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}[/tex].
[tex]\Rightarrow S_{DAEL }=S_{\triangle DAB }- S_{\triangle BEL}=3x-\frac{3x}{4}=\frac{9x}{4};[/tex] similarly [tex]\triangle DFL\approx \triangle DBC\Rightarrow S_{\triangle DLF}=\frac{7x}{4}[/tex]
[tex]\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 ;[/tex]
[tex]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}[/tex].

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