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If A is the area and 2s is the sum of th...

If `A` is the area and `2s` is the sum of the sides of a triangle, then `Alt=(s^2)/4` (b) `Alt=(s^2)/(3sqrt(3))` `2RsinAsinBsinC` (d) `non eoft h e s e`

A

`Ale (s^2)/(4)`

B

`Ale(s^2)/(3sqrt(3))`

C

`Alt (s^2)/(sqrt(3)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A, B
We have
`2s = a + b + c`
`A^(2) = s (s - a) (s - b)(s - c)`
Now,
`A.M. ge G.M`
`implies (s + (s - a) + (s - b) + (s - c))/(4) ge [s(s-a)(s-b)(s-c)]^(1//4)`
`implies (4s - 2s)/(4) ge [A^(2)]^(1//4)`
`implies 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)`
`implies (s)/(3) ge [(A^(2))/(s)]^(1//3)`
or `(A^(2))/(s) le (s^(3))/(27)`
or `A le (s^(2))/(3sqrt(3))`
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