Question

If $\Delta$ denotes the area of any triangle and $s$ its semi-perimeter, then $\Delta <\frac{s^2}{k}$

Answer 1

Let $a,b,c$ be the sides of the triangle.

Clearly, $s=\frac{a+b+c}{2}$ and so $s,s-a,s-b,s-c$ will be positive.

$\because$ For positive quantities, $A.M.>G.M.$ $[\because s,s-a,s-b,s-c$ are not all equal $]$

$\therefore \frac{s+(s-a)+(s-b)+(s-c)}{4}>\left[s(s-a)(s-b)(s-c)\right]$

$\Rightarrow \frac{4s-(a+b+c)}{4}>{\left({\Delta }^2\right)}^{\frac{1}{4}}\Rightarrow \frac{2s}{4}>{\Delta }^{\frac{1}{2}}[\because a+b+c=2s]$

$\Rightarrow \frac{s^2}{4}>\Delta$ $[\because$ both $s$ and $\Delta$ are positive, squaring will not affect the inequality $]$