Question 814612
Heron's Formula for area is

A = sqrt(s(s-a)(s-b)(s-c)  where s = (a + b + c)/2 with a,b,c being side lengths.
Since the side ratios are 17:10:9 we can write
rs = r(17+10+9)/2 = r*(36)/2 = r*18 = 18r   where r is proportion factor
A = {{{sqrt(18r(18r-17r)(18r-10r)(18r-9r))}}}
A = {{{sqrt(18r(r)(8r)(9r))}}}   
A = {{{sqrt(18(8)(9)r^4)}}}
Factoring the numbers we get
A = {{{sqrt(2*3*3 * 2*2*2 * 3*3 r^4)}}}
A = {{{sqrt( 2^4 * 3^4 * r^4 )}}}
A = {{{2^2 * 3^2 * r^2}}}
A = {{{4 * 9 * r^2}}}
A = {{{36 * r^2}}}
Since A = 576, and solving for r
576 = {{{36 * r^2}}}
Divide each side by 36
16 = {{{r^2}}}
So r = 4
This means that our side lengths are 4*17 , 4*10 , 4*9

The side lengths of our triangle are 68 , 40 , 36