ΔABC and ΔDEF are similar to each other. If the ratio of side AB to side DE is $(\sqrt{2} + 1) : \sqrt{3}$, then the ratio of area of ΔABC to that of ΔDEF is:

1. $(3 + 2\sqrt{2}) : 3$

2. $1 : (9 - 6\sqrt{2})$

3. $(9 - 6\sqrt{2}) : 2$

4. $(3 - 2\sqrt{2}) : 3$

Given,

ΔABC and ΔDEF are similar to each other.

Since the triangles are similar, the ratios of Areas of triangles is equal to squares of corresponding sides.

$\Rightarrow \dfrac{\text{Area of triangle ABC}}{\text{Area of triangle DEF}} = \Big(\dfrac{(\sqrt{2} + 1)}{\sqrt{3}}\Big)^2 \\[1em] = \dfrac{(\sqrt{2} + 1)^2}{(\sqrt{3})^2} \\[1em] = \dfrac{(\sqrt{2})^2 + (1)^2 + 2 \times \sqrt{2} \times 1}{3} \\[1em] = \dfrac{2 + 1 + 2\sqrt{2}}{3} \\[1em] = \dfrac{3 + 2\sqrt{2}}{3}.$

Area of triangle ABC : Area of triangle DEF = $(3 + 2\sqrt{2}) : 3$.

Hence, option 1 is the correct option.