girishbtg wrote:

When the areas of two similar triangles are in ratio 1:2, their corresponding sides are in the ratio equivalent to the square root of 1:2, which is 1:√2.

Thus, we have s:S::1:√2, remember, product of extremes = product of means, a rule of proportionality

∴ S = [spoiler]s √2


D[/spoiler]

Alternative method: the "note: figure not drawn to scale" is a clue - it means that the shape of the triangles is not important - the answer is correct regardless of how the triangles are formed. Specifically, you don't know the value of the angles x, y and z, but the values do not matter, for the right answer is right for any value of x, y and z you choose.

Liberation! You can then choose the values that make your life easier - choose x=90, y and z as 45 degrees, transforming the triangles from generic similar triangles into isosceles triangles with a base and height of s and S.
Area of small triangle: 1/2*s^2
Area of Large triangle : 1/2*S^2.
according to the question, area of large is twice area of small, so S^2 / 2 = 2*s^2/2
S^2 / 2 = s^2
S^2 = 2s^2
Take the square root to find:
S = s*√2.

Shortcut formulae when it involves similar triangles is -

The ratio of the areas of 2 similar triangles = square of the ratio of the corresponding sides. This implies -

2 S^2
- = -----
1 s^2

=> S = sqrt(2).s