Question

1)cho hình thang ABCD(AB//CD),O là giao điểm 2 đường chéo

CMR: Saob+Scod>=1/2Sabcd

 

Answer 1

Ta có : \(\frac{S_{BOC}}{S_{COD}}=\frac{OB}{OD}\); \(\frac{S_{AOB}}{S_{AOD}}=\frac{OB}{OD}\)

\(\Rightarrow\frac{S_{BOC}}{S_{COD}}=\frac{S_{AOB}}{S_{AOD}}\Rightarrow S_{BOC}.S_{AOD}=S_{AOB}.S_{COD}\)

Lại có : \(S_{ABCD}=S_{AOB}+S_{COD}+\left(S_{BOC}+S_{AOD}\right)=S_{AOB}+S_{COD}+2\sqrt{S_{BOC}.S_{AOD}}=S_{AOB}+S_{COD}+2\sqrt{S_{AOB}.S_{COD}}=\left(\sqrt{S_{AOB}}+\sqrt{S_{COD}}\right)^2\)( Vì \(S_{BOC}=S_{AOD}\))

Mặt khác : \(S_{ABCD}=\left(\sqrt{S_{AOB}}+\sqrt{S_{COD}}\right)^2=\left(1.\sqrt{S_{AOB}}+1.\sqrt{S_{COD}}\right)^2\le2\left(S_{AOB}+S_{COD}\right)\Rightarrow S_{AOB}+S_{COD}\ge\frac{1}{2}.S_{ABCD}\)(ĐPCM)