If $Δ$ denotes the area of any triangle and s its semi - perimeter, then

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Solution

The correct option is C

$Let a, b, c be the sides of the triangle. Clearly , s = a+b+c/2 and so s, s - a, s - b, s - c will be positive ∵ For positive quantities, A M \geq G M [∴ s, s - a, s - b, s - c are not all equal ∴ \frac{s + (s - a) + (s + b) + (s - c)}{4} > [s (s - a) (s - b) (s - c)]^{1 / 4} o r \frac{4 s - (a + b + c)}{4} > (Δ^{2})^{1 / 4} o r \frac{2 s}{4} > Δ^{1 / 2} [∵ a + b + c = 2 s] o r \frac{s^{2}}{4} > Δ [∵ both s and D are positive, squaring will not affect inequality] i . e ., Δ < \frac{s^{2}}{4}$

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If Δ denotes the area of any triangle and s its semi - perimeter, then

If $Δ$ denotes the area of any triangle and s its semi - perimeter, then