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Question Number 58876 by mr W last updated on 01/May/19

Commented by math1967 last updated on 01/May/19

Is it (1/3) part sir?

$${Is}\:{it}\:\frac{\mathrm{1}}{\mathrm{3}}\:{part}\:{sir}? \\ $$

Commented by mr W last updated on 01/May/19

correct sir!

$${correct}\:{sir}! \\ $$

Commented by math1967 last updated on 01/May/19

△AOB∼△EOD ∴((BO)/(DO))=((AB)/(DE))=(2/1)  Again ((Ar△AOB)/(Ar△AOD))=((BO)/(DO))=(2/1)  ∴△AOB:△DAB=2:3  ∴△AOB=(2/3)△DAB=(2/3)×(1/2)ABCD                                                  =(1/3)ABCD

$$\bigtriangleup{AOB}\sim\bigtriangleup{EOD}\:\therefore\frac{{BO}}{{DO}}=\frac{{AB}}{{DE}}=\frac{\mathrm{2}}{\mathrm{1}} \\ $$$${Again}\:\frac{{Ar}\bigtriangleup{AOB}}{{Ar}\bigtriangleup{AOD}}=\frac{{BO}}{{DO}}=\frac{\mathrm{2}}{\mathrm{1}} \\ $$$$\therefore\bigtriangleup{AOB}:\bigtriangleup{DAB}=\mathrm{2}:\mathrm{3} \\ $$$$\therefore\bigtriangleup{AOB}=\frac{\mathrm{2}}{\mathrm{3}}\bigtriangleup{DAB}=\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{1}}{\mathrm{2}}{ABCD}\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{3}}{ABCD} \\ $$

Commented by math1967 last updated on 01/May/19

Answered by tanmay last updated on 01/May/19

tanθ=((a/2)/a)=(1/2)  the eq of st lines which enclose the shaded area  (1) x axis   (2) (x/a)+(y/a)=1(3)  y=tan(90−θ)x  y=cotθx→y=2x  now solve y=2x and x+y=a  3x=a  x=(a/3) and y=((2a)/3)  the shaded triangle  say ABC  point A(0,0)  B(a,0)  C((a/3),((2a)/3))  area of △ABC=(1/2)×a×((2a)/3)=(a^2 /3)  area of shaded triangle=(1/3)(area of square)

$${tan}\theta=\frac{\frac{{a}}{\mathrm{2}}}{{a}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${the}\:{eq}\:{of}\:{st}\:{lines}\:{which}\:{enclose}\:{the}\:{shaded}\:{area} \\ $$$$\left(\mathrm{1}\right)\:{x}\:{axis}\:\:\:\left(\mathrm{2}\right)\:\frac{{x}}{{a}}+\frac{{y}}{{a}}=\mathrm{1}\left(\mathrm{3}\right)\:\:{y}={tan}\left(\mathrm{90}−\theta\right){x} \\ $$$${y}={cot}\theta{x}\rightarrow{y}=\mathrm{2}{x} \\ $$$${now}\:{solve}\:{y}=\mathrm{2}{x}\:{and}\:{x}+{y}={a} \\ $$$$\mathrm{3}{x}={a}\:\:{x}=\frac{{a}}{\mathrm{3}}\:{and}\:{y}=\frac{\mathrm{2}{a}}{\mathrm{3}} \\ $$$${the}\:{shaded}\:{triangle}\:\:{say}\:{ABC} \\ $$$${point}\:{A}\left(\mathrm{0},\mathrm{0}\right)\:\:{B}\left({a},\mathrm{0}\right)\:\:{C}\left(\frac{{a}}{\mathrm{3}},\frac{\mathrm{2}{a}}{\mathrm{3}}\right) \\ $$$${area}\:{of}\:\bigtriangleup{ABC}=\frac{\mathrm{1}}{\mathrm{2}}×{a}×\frac{\mathrm{2}{a}}{\mathrm{3}}=\frac{{a}^{\mathrm{2}} }{\mathrm{3}} \\ $$$${area}\:{of}\:{shaded}\:{triangle}=\frac{\mathrm{1}}{\mathrm{3}}\left({area}\:{of}\:{square}\right) \\ $$$$ \\ $$$$ \\ $$

Commented by tanmay last updated on 01/May/19

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