let us assume we divide string at length x such that x is bigger part i.e x>1/2 and smaller part is 1-x ,further we divide x into 2parts of ratio 4:3 means 1 part is 4x/7 and second part is 3x/7 so far we have establish the value of each side (a,b,c) in term of x (4x/7,3x/7,1-x)
now for triangle to form following 2 property must satisfies a+b>c and |a-b|<c for any side (a,b,c) now we have to permute a b c as per above to find value of x
1)a=4x/7 b=3x/7,c=1-x ----- x belong to (1/2 ,7/8)
2)a=4x/7 b=1-x ,c=3x/7 ----- x < 7/6 futile because x cannot be greater than 1
3)a=3x/7 b=1-x ,c=4x/7 ----- x belong to (1/2 ,7/6)
4)a=3x/7 b=4x/7 ,c=1-x ----- x belong to (1/2 ,7/8)
5)a=1-x b=4x/7 ,c=3x/7 ----- x < 7/8
6)a=1-x b=3x/7 ,c=4x/7 ----- x < 7/6 futile because x cannot be greater than 1
so finally x should be between 1/2 and 7/8 inorder to form triangle hence it can take (7/8-1/2) =3/8 values therefore
probabilty of forming a triangle is 3/8(favourable values)/ 1/2(total values)=3/4
probability of not forming a triangle will be 1-3/4 = 1/4