If `A` is the area and `2s` is the sum of the sides of a triangle, then

Options

A. `A le (s^(2))/(4)`
B. `A le (s^(2))/(3sqrt3)`
C. `A lt (s^(2))/(sqrt3)`
D. none of these

Answer

A, B

Explanation

We have `2s = a + b + c, A^(2) = s (s-a) (s-b) (s-c)`
Now, `A.M. ge G.M.`
`rArr (s+(s-a) +(s-b)+(s-c))/(4)`
`ge [s (s-a) (s-b) (s-c)]^(1//4)`
`rArr (4s-2s)/(4) ge [A^(2)]^(1//4)`
`rArr ((s)/(2)) ge A^(1//2) or A le ((s^(2))/(4))`
Also, `((s-a) + (s-b) + (s-c))/(3)`
`ge [(s-a) (s-b) (s-c)]^(1//3)`
or `(s)/(3) ge [(A^(2))/(s)]^(1//3) or (A^(2))/(s) le (s^(3))/(27) or A le (s^(2))/(3sqrt3)`