1.

In any triangle, the minimum value of `r_1r_2r_3//r^3`is equal to1 (b)9 (c) 27(d) none of these

Answer» Here, we will use the property G.M. is always greater than or equal to H.M.
`:. (r_1/r* r_2/r* r_3/r )^(1/3) ge 3/(r/r_1+ r/r_2+ r/r_3)->(1)`
Now, `r = Delta/s, r_1 = Delta/(s-a),r_2 = Delta/(s-b),r_3 = Delta/(s-c)`
`:. r/r_1 = (s-a)/s, r/r_2 = (s-b)/s,r/r_3 = (s-c)/s`
`:. r/r_1+r/r_2+r/r_3 = 1/s[s-a+s-b+s-c] = 1/s[3s-(a+b+c)]`
`= 1/s[3s-2s] = 1`
`:. r/r_1+r/r_2+r/r_3 = 1`
So, `(1)` becomes,
`=> (r_1/r* r_2/r* r_3/r )^(1/3) ge 3/1`
`=>(r_1/r* r_2/r* r_3/r ) ge 3^3`
`=>(r_1r_2r_3)/r^3 ge 27`
So, minimum value of `(r_1r_2r_3)/r^3` will be `27`.


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