Đáp án:

$S_{\Delta CID}=162(cm^2)$

Giải thích các bước giải:

Kẻ $AH\perp DC; AE,CF \perp BD$

$S_{\Delta ABD}=\dfrac{1}{2}AH.AB\\ S_{\Delta CBD}=\dfrac{1}{2}AH.DC\\ \Rightarrow \dfrac{S_{\Delta ABD}}{S_{\Delta CBD}}=\dfrac{AB}{DC}=\dfrac{2}{3}(1)\\ S_{\Delta ABD}=\dfrac{1}{2}AE.BD\\ S_{\Delta ABD}=\dfrac{1}{2}CF.BD\\ \Rightarrow  \dfrac{S_{\Delta ABD}}{S_{\Delta CBD}}=\dfrac{AE}{CF}(2)\\ (1)(2) \Rightarrow  \dfrac{AE}{CF}=\dfrac{2}{3}\\ S_{\Delta CID}=\dfrac{1}{2}CF.DI\\ S_{\Delta AID}=\dfrac{1}{2}AE.DI\\ \Rightarrow \dfrac{S_{\Delta CID}}{S_{\Delta AID}}=\dfrac{CF}{AE}=\dfrac{3}{2}\\ S_{\Delta ADC}=S_{\Delta AID}+S_{\Delta CID}=\dfrac{2}{3}S_{\Delta CID}+S_{\Delta CID}=\dfrac{5}{3}S_{\Delta CID}\\ S_{\Delta ADC}=\dfrac{1}{2}AH.DC\\ S_{\Delta ABC}=\dfrac{1}{2}AH.AB\\ \Rightarrow \dfrac{S_{\Delta ADC}}{S_{\Delta ABC}}=\dfrac{DC}{AB}=\dfrac{3}{2}\\ S_{ABCD}\\ =S_{\Delta ADC}+S_{\Delta ABC}\\ =S_{\Delta ADC}+\dfrac{2}{3}S_{\Delta ADC}\\ =\dfrac{5}{3}S_{\Delta ADC}\\ =\dfrac{5}{3}.\dfrac{5}{3}S_{\Delta CID}\\ =\dfrac{25}{9}S_{\Delta CID}\\ \Rightarrow S_{\Delta CID}=\dfrac{9}{25}S_{ABCD}=\dfrac{9}{25}.450=162(cm^2)$