Explanation:

Here's the solution explained step by step in clear points:

- An equilateral triangle can be cut at its corners (each cut is parallel to the opposite side) to form a regular hexagon.

- Let the side of the original triangle be \(a\).

- The largest hexagon inscribed in an equilateral triangle has each of its sides one-third the side of the triangle.

- Area of the equilateral triangle: \(\frac{\sqrt{3}}{4}a^2\)

- Side of the hexagon: Let’s call it \(x\). \(x = \frac{a}{3}\)

- Area of regular hexagon:

$$6 \times \frac{\sqrt{3}}{4}x^2 = \frac{3\sqrt{3}}{2}x^2$$

Substitute \(x = \frac{a}{3}\):

$$

= \frac{3\sqrt{3}}{2}\left(\frac{a^2}{9}\right) = \frac{\sqrt{3}}{6}a^2

$$

- Ratio of triangle area to hexagon area:

$$

\frac{\frac{\sqrt{3}}{4}a^2}{\frac{\sqrt{3}}{6}a^2} = \frac{6}{4} = \frac{3}{2}

$$

- Among the options, Option 2: 3:2 matches.

Correct Answer: Option 2: 3 : 2