If A is the area and 2s is the sum of 3 sides | vedclass

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(A) We have $2s = a + b + c$ and the area $A$ is given by Heron's formula: $A^2 = s(s - a)(s - b)(s - c)$.By the Arithmetic Mean-Geometric Mean $(AM \

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$A \le \frac{s^2}{3\sqrt{3}}$

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(A) We have $2s = a + b + c$ and the area $A$ is given by Heron's formula: $A^2 = s(s - a)(s - b)(s - c)$.By the Arithmetic Mean-Geometric Mean $(AM \ge GM)$ inequality for the positive terms $(s-a), (s-b), (s-c)$:$\frac{(s-a) + (s-b) + (s-c)}{3} \ge \sqrt[3]{(s-a)(s-b)(s-c)}$Substituting $a+b+c = 2s$:$\frac{3s - (a+b+c)}{3} \ge \sqrt[3]{\frac{A^2}{s}}$$\frac{3s - 2s}{3} \ge \sqrt[3]{\frac{A^2}{s}}$$\frac{s}{3} \ge \sqrt[3]{\frac{A^2}{s}}$Cubing both sides:$\frac{s^3}{27} \ge \frac{A^2}{s}$$A^2 \le \frac{s^4}{27}$Taking the square root:$A \le \frac{s^2}{3\sqrt{3}}$.

If $A$ is the area and $2s$ is the sum of $3$ sides of a triangle,then:

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