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Question Number 210050 by Batmath last updated on 29/Jul/24

Answered by mr W last updated on 29/Jul/24

Commented by mr W last updated on 29/Jul/24

$$\frac{2}{3}\times (6+\frac{3x}{2})=3x$$$$\Rightarrow x=2$$

Answered by A5T last updated on 29/Jul/24

Commented by A5T last updated on 29/Jul/24

$$\frac{AD}{DB}\times \frac{BC}{CG}\times \frac{GE}{EA}=1\Rightarrow \frac{AD}{DB}\times 2\times \frac{3x}{12}=1\Rightarrow \frac{AD}{DB}=\frac{2}{x}$$$$\frac{CG}{GB}\times \frac{BA}{AD}\times \frac{DE}{EC}=1\Rightarrow 1\times \frac{2+x}{2}\times \frac{1}{2}=1$$$$\Rightarrow 2+x=4\Rightarrow x=2$$

Answered by A5T last updated on 29/Jul/24

Commented by A5T last updated on 29/Jul/24

$$Gand{G}^{\prime }arethecentroidsof△ABCand△GBC$$$$respectively\Rightarrow \frac{{GG}^{\prime }}{x}=2and\frac{12}{{GG}^{\prime }+x}=2$$$$\Rightarrow \frac{12}{2x+x}=2\Rightarrow x=2$$

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