1.

If area of a triangle is 2 sq. units, then find the value of theproduct of the arithmetic mean of the lengths of the sides of a triangle andharmonic mean of the lengths of the altitudes of the triangle.

Answer» Let `a,b and c` are the sides and `h_1,h_2 and h_3` are the lengths of the altitudes.
Then,
`ah_1 = bh_2 = ch_3 = 2Delta`
`=>1/h_1+1/h_2+1/h_3 = (a+b+c)/(2Delta)`
`=>2Delta = (a+b+c)/(1/h_1+1/h_2+1/h_3)`
`=>2Delta = (a+b+c)/3**3/(1/h_1+1/h_2+1/h_3)`
Now, it is given that `Delta = 2` square units
`:. 2*2 = (a+b+c)/3**3/(1/h_1+1/h_2+1/h_3)`
`=>(a+b+c)/3**3/(1/h_1+1/h_2+1/h_3) = 4`
Here, `(a+b+c)/3` is A.M. of the sides and `3/(1/h_1+1/h_2+1/h_3)` is the H.M. of the altitudes.
Therefore, the required product is `4`.


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