Let the semi-perimeter of a triangle bes. Then, the maximum area of the triangle is

• A)

  

• B)

  

• C)

  

• D)

  

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{area} \mathrm{of} \mathrm{fixed} \mathrm{triangle} \mathrm{is} \mathrm{maximum} \mathrm{when} \mathrm{it} \mathrm{is} \mathrm{an} \mathrm{equilateral} \mathrm{triangle}. \\ \mathrm{s}=\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{2} \\ \mathrm{All} \mathrm{sides} \mathrm{of} \mathrm{equilateral} \mathrm{triangle} \mathrm{are} \mathrm{equal}. \\ \Rightarrow \mathrm{s}=\frac{\mathrm{a}+\mathrm{a}+\mathrm{a}}{2} \\ \Leftarrow \mathrm{a}=\frac{2\mathrm{s}}{3} \\ \mathrm{Area} =\frac{\sqrt{3}}{4}{\mathrm{a}}^2 \\ =\frac{\sqrt{3}}{4}{\left(\frac{2\mathrm{s}}{3}\right)}^2 \\ =\frac{\sqrt{3}}{4}\frac{4{\mathrm{s}}^2}{9} \\ =\frac{{\mathrm{s}}^2}{3\sqrt{3}}....\mathrm{Option}\left(\mathrm{A}\right) \end{gathered}$$

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