Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$\text{Let a, b, c be the sides of the triangle.}\text{Clearly , s = a+b+c/2 and so}\text{s, s - a, s - b, s - c will be positive}\because \text{For positive quantities,}AM\ge GM[\therefore s,s-a,s-b,s-c\text{are not all equal}\therefore \frac{s+(s-a)+(s+b)+(s-c)}{4}>[s(s-a)(s-b)(s-c){]}^{1/4}or\frac{4s-(a+b+c)}{4}>({\Delta }^2{)}^{1/4}or\frac{2s}{4}>{\Delta }^{1/2}[\because a+b+c=2s]or\frac{s^2}{4}>\Delta [\because \text{both s and D are positive, squaring will not affect inequality}]i.e.,\Delta <\frac{s^2}{4}$