Question Number 63194 by lalitchand last updated on 30/Jun/19
Answered by mr W last updated on 30/Jun/19
$$\frac{\Delta{AQR}}{\Delta{ABC}}=\left(\frac{{AQ}}{{AB}}\right)^{\mathrm{2}} =\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{2}} =\frac{\mathrm{4}}{\mathrm{9}} \\ $$$$\frac{\Delta{APS}}{\Delta{ABC}}=\left(\frac{{AP}}{{AB}}\right)^{\mathrm{2}} =\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{9}} \\ $$$$\frac{\Box{PQRS}}{\Delta{ABC}}=\frac{\Delta{AQR}−\Delta{APS}}{\Delta{ABC}}=\frac{\Delta{AQR}}{\Delta{ABC}}−\frac{\Delta{APS}}{\Delta{ABC}}=\frac{\mathrm{4}}{\mathrm{9}}−\frac{\mathrm{1}}{\mathrm{9}}=\frac{\mathrm{3}}{\mathrm{9}}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$