Assuming H,h as Height of triangle

From Given Info
Twice the area of small triangle = Area of large triangle
Area of small triangle = (1/2) (s * h)
Area of Large triangle = (1/2) (S * H)

(1/2) (s * h) (2) = (1/2) (S * H)

(s *h) = (S * H) (1/2) -------->(1)

Now since the angles are equal the similar triangles sides/Height ratio would also be the same

S= x * s ----------->(2)

H= x * h ----------->(3)

Substituting (2) and (3) in (1)

(s*h) = (x^2)(s * h) (1/2)

x = sqrt(2)

sub this in eqn (2)

S = sqrt(2) * s