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Question Number 1592 by Rasheed Soomro last updated on 23/Aug/15

$$\mathrm{I}\mathrm{have}\mathrm{a}\mathrm{loop}\mathrm{of}\mathrm{string}\mathrm{of}\mathrm{length}\left(\mathrm{perimeter}\right)\mathrm{p}\mathrm{units}.$$$$\mathrm{I}\mathrm{want}\mathrm{to}\mathrm{make}\mathrm{a}\mathrm{triangle}\mathrm{of}\mathrm{largest}\mathrm{area}\mathrm{from}\mathrm{the}$$$$\mathrm{loop}.\mathrm{What}\mathrm{will}\mathrm{be}\mathrm{the}\mathrm{dimensions}\mathrm{of}\mathrm{that}\mathrm{triangle}?$$

Commented by 123456 last updated on 23/Aug/15

$$x+y+z=p$$$$s=p/2$$$$\{\begin{array}{l}\mid x-y\mid ⩽z⩽x+y\\ \mid x-z\mid ⩽y⩽x+z\\ \mid y-z\mid ⩽x⩽y+z\end{array}$$$$\mathrm{S}(x,y,z)=\sqrt{s(s-x)(s-y)(s-z)}$$

Answered by prakash jain last updated on 10/Dec/15

$$x=y=z=\frac{p}{3}$$$$\mathrm{among}\mathrm{all}\mathrm{triangle}\mathrm{of}\mathrm{same}\mathrm{perimeter}$$$$\mathrm{equilateral}\mathrm{triangle}\mathrm{has}\mathrm{largest}\mathrm{area}.$$

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