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Question Number 47010 by ajfour last updated on 03/Nov/18

Commented by ajfour last updated on 03/Nov/18

$$Findmaximumrectangular$$$$areaand\mathit{l}\mathrm{\&}\mathit{h}\left(then\right),interms$$$$ofsidesa,b,cof△ABC.$$$$\left(withQRalongBC\right)$$

Answered by ajfour last updated on 04/Nov/18

$$area(△ABC)={△}_0$$$$\frac{Ar(△APS)}{{△}_0}={\left(\frac{l}{a}\right)}^2$$$$\frac{Ar(△PBQ)+Ar(△SRC)}{{△}_9}={\left(\frac{a-l}{a}\right)}^2$$$$\Rightarrow lh={△}_0[1-{\left(\frac{l}{a}\right)}^2-{(1-\frac{l}{a})}^2]$$$$=2{△}_0\left(\frac{l}{a}\right)(1-\frac{l}{a})$$$$\Rightarrow max\left(lh\right)when\frac{l}{a}=\frac{1}{2}$$$$max\left(PQRS\right)=\frac{{△}_0}{2},l=\frac{a}{2}$$$$h=\frac{altitudeof△ABC}{2}.$$

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