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Question Number 83694 by naka3546 last updated on 05/Mar/20

Commented by naka3546 last updated on 05/Mar/20

$$DE=EF=FC,$$$$BG=2CG,$$$$ABCDisasquare.$$$$\frac{\left[BMN\right]}{\left[ABCD\right]}=?$$$$\left(withoutTrigonometryorMenelause\right)$$

Commented by john santu last updated on 05/Mar/20

$$\mathrm{what}\mathrm{the}\mathrm{meaning}\left[\mathrm{BMN}\right]?$$$$\mathrm{area}?$$

Answered by MJS last updated on 05/Mar/20

$$A=\left(\begin{array}{c}0\\ 3\end{array}\right)B=\left(\begin{array}{c}3\\ 3\end{array}\right)C=\left(\begin{array}{c}3\\ 0\end{array}\right)D=\left(\begin{array}{c}0\\ 0\end{array}\right)$$$$E=\left(\begin{array}{c}1\\ 0\end{array}\right)F=\left(\begin{array}{c}2\\ 0\end{array}\right)G=\left(\begin{array}{c}3\\ 1\end{array}\right)$$$$\mathrm{line}DG:y=\frac{1}{3}x$$$$\mathrm{line}EB:y=\frac{3}{2}x-\frac{3}{2}$$$$\mathrm{line}FB:y=3x-6$$$$DG\cap EB=M=\left(\begin{array}{c}9/7\\ 3/7\end{array}\right)$$$$DG\cap FB=N=\left(\begin{array}{c}9/4\\ 3/4\end{array}\right)$$$$\mid BM\mid =\frac{6\sqrt{13}}{7}$$$$\mid BN\mid =\frac{3\sqrt{10}}{4}$$$$\mid MN\mid =\frac{9\sqrt{10}}{28}$$$$\mathrm{area}\left(abc\right)=\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a+b+c)(a+b-c)}$$$$\Rightarrow \mathrm{area}\left(BMN\right)=\frac{27}{28}$$$$\mathrm{area}\left(ABCD\right)=9$$$$\Rightarrow \frac{\mathrm{area}\left(BMN\right)}{\mathrm{area}\left(ABCD\right)}=\frac{3}{28}$$

Answered by mr W last updated on 05/Mar/20

Commented by mr W last updated on 05/Mar/20

$$ABCDisrectangle,{mustn}^{\prime }tbesquare.$$$$\frac{EM}{MB}=\frac{PE}{BG}=\frac{1}{3}\times \frac{CG}{BG}=\frac{1}{3}\times \frac{1}{2}=\frac{1}{6}$$$$\Rightarrow \frac{MB}{EB}=\frac{6}{7}$$$$\frac{FN}{NB}=\frac{QF}{BG}=\frac{2}{3}\times \frac{CG}{BG}=\frac{2}{3}\times \frac{1}{2}=\frac{1}{3}$$$$\Rightarrow \frac{NB}{FB}=\frac{3}{4}$$$$\frac{\left[BMN\right]}{\left[BEF\right]}=\frac{MB}{EB}\times \frac{NB}{FB}=\frac{6}{7}\times \frac{3}{4}=\frac{18}{28}$$$$\frac{\left[BEF\right]}{\left[ABCD\right]}=\frac{1}{6}$$$$\Rightarrow \frac{\left[BMN\right]}{\left[ABCD\right]}=\frac{18}{28}\times \frac{1}{6}=\frac{3}{28}$$

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