Ratio

When placing the hypotenuses of each pair of right triangles together, we get two rectangles similar to each other. Thus the ratio of their areas is the square of the ratio of their diagonals: $\Big(\frac{8}{6}\Big)^2 = \frac{16}{9} = 1.\overline{7}$

let (a) is the height of the two triangles (b) is the base of shaded triangle and (c) is the base of the unshaded triangle , 64=a^2+b^2 , 36=a^2+c^2 , b^2-c^2=28 , b+c=10 , b=10-c , (10-c)-c^2=28 , 100+c^2-20 c-c^2=28 , c=(100-28)/20
=3.6 , b=6.4 , area of shaded region A1=2
(0.5 a b) , area of unshaded region A2=a*c , A1/A2=b/c =1.777778