Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

Question Number 103774 by dw last updated on 17/Jul/20

Answered by mr W last updated on 17/Jul/20

$$\mathit{M}\mathit{E}\mathit{T}\mathit{H}\mathit{O}\mathit{D}\mathit{I}$$$$\overrightarrow{AB}=\mathit{a}$$$$\overrightarrow{BC}=\mathit{b}$$$$\overrightarrow{CA}=\mathit{c}$$$$\mathit{a}+\mathit{b}+\mathit{c}=0$$$$$$$$AH=\mathit{a}+\frac{1}{9}\mathit{b}$$$$BG=-\mathit{a}-\frac{1}{3}\mathit{c}$$$$CP=-\mathit{b}-\frac{5}{6}\mathit{a}$$$$BE=\lambda BG=-\lambda \mathit{a}-\frac{\lambda }{3}\mathit{c}$$$$AE=\mu AH=\mu \mathit{a}+\frac{\mu }{9}\mathit{b}=\mathit{a}-\lambda \mathit{a}-\frac{\lambda }{3}\mathit{c}$$$$\mu \mathit{a}+\frac{\mu }{9}\mathit{b}=(1-\lambda )\mathit{a}+\frac{\lambda }{3}(\mathit{a}+\mathit{b})$$$$\mu \mathit{a}+\frac{\mu }{9}\mathit{b}=(1-\frac{2}{3}\lambda )\mathit{a}+\frac{\lambda }{3}\mathit{b}$$$$\Rightarrow \mu =3\lambda$$$$\Rightarrow \mu =3\lambda =1-\frac{2}{3}\lambda$$$$\Rightarrow \lambda =\frac{3}{11}\Rightarrow \mu =\frac{9}{11}$$$$$$$$AF=\xi AH=\xi \mathit{a}+\frac{\xi }{9}\mathit{b}$$$$CF=\nu CP=-\nu \mathit{a}-\frac{5\nu }{6}\mathit{b}=CA+AF$$$$-\nu \mathit{b}-\frac{5\nu }{6}\mathit{a}=-\mathit{a}-\mathit{b}+\xi \mathit{a}+\frac{\xi }{9}\mathit{b}$$$$-\frac{5\nu }{6}\mathit{b}-\nu \mathit{a}=(\xi -1)\mathit{a}+(\frac{\xi }{9}-1)\mathit{b}$$$$\Rightarrow -\nu =\frac{\xi }{9}-1$$$$\Rightarrow -\frac{5\nu }{6}=\xi -1$$$$\Rightarrow \frac{5}{6}(\frac{\xi }{9}-1)=\xi -1$$$$\Rightarrow \xi =\frac{9}{49}$$$$\Rightarrow \nu =\frac{48}{49}$$$$$$$$BD=\eta BG=-\eta \mathit{a}+\frac{\eta }{3}(\mathit{a}+\mathit{b})=-\frac{2\eta }{3}\mathit{a}+\frac{\eta }{3}\mathit{b}$$$$CD=ϵCP=-ϵ\mathit{b}-\frac{5ϵ}{6}\mathit{a}=CB+BD$$$$-ϵ\mathit{b}-\frac{5ϵ}{6}\mathit{a}=-\frac{2\eta }{3}\mathit{a}+(\frac{\eta }{3}-1)\mathit{b}$$$$\Rightarrow -\frac{5ϵ}{6}=-\frac{2\eta }{3}\Rightarrow 5ϵ=4\eta$$$$\Rightarrow -ϵ=\frac{\eta }{3}-1$$$$\Rightarrow \eta =\frac{15}{17}$$$$\Rightarrow ϵ=\frac{12}{17}$$$$$$$$ED=BD-BE=(\eta -\lambda )BG=$$$$=(\frac{15}{17}-\frac{3}{11})BG=\frac{114}{187}BG$$$$EG=BG-BE=(1-\lambda )BG=\frac{8}{11}BG$$$$\frac{ED}{EG}=\frac{114}{187}\times \frac{11}{8}=\frac{57}{68}$$$$EF=AE-AF=(\mu -\xi )AH$$$$=(\frac{9}{11}-\frac{9}{49})AH=\frac{342}{539}AH$$$$\frac{EF}{EA}=\frac{342}{539}\times \frac{1}{\mu }=\frac{342}{539}\times \frac{11}{9}=\frac{38}{49}$$$$$$$$S_{DEF}=\frac{EF}{EA}\times \frac{ED}{EG}\times S_{AEG}$$$$=\frac{EF}{EA}\times \frac{ED}{EG}\times \frac{EG}{BG}\times S_{ABG}$$$$=\frac{EF}{EA}\times \frac{ED}{EG}\times \frac{EG}{BG}\times \frac{AG}{AC}\times S_{ABC}$$$$=\frac{38}{49}\times \frac{57}{68}\times \frac{8}{11}\times \frac{1}{3}\times S_{ABC}$$$$=\frac{1444}{9163}\times S_{ABC}$$$$\Rightarrow \frac{S_{DEF}}{S_{ABC}}=\frac{1444}{9163}\approx 0.1576$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com