To find the area of the plank, we need to determine the length and width of the rectangular plank.

Let's consider a square with side length s. The diagonal of the square divides it into two congruent right triangles. According to the Pythagorean theorem, the length of the diagonal (d) can be calculated as:

d = √(s^2 + s^2) = √(2s^2) = s√2

Given that the width of the plank is √2 meters, it means that the length of one side of the rectangular plank is √2 meters.

To determine the length of the other side of the rectangular plank, we need to consider the two possible positions of the plank.

Position 1: The diagonal of the square divides the plank into two congruent right triangles. The width of the plank (√2 meters) is equal to the length of one side of each right triangle. Therefore, the length of the other side of each right triangle is s (the side length of the square). This means that the length of the rectangular plank is s.

Position 2: The diagonal of the square is the length of the plank. In this case, the width of the plank (√2 meters) is equal to the length of one side of each right triangle. Therefore, the length of the other side of each right triangle is s√2. This means that the length of the rectangular plank is s√2.

So, in both positions, the length of the rectangular plank is either s or s√2.

Since the width of the plank is √2 meters, the length of the plank must be s√2. Therefore, the area of the plank is:

Area = length × width = (s√2) × (√2) = 2s

We know that the area of the square is given by s^2. So, the area of the plank is 2s.

Given that the side length of the square is √2 meters, we can substitute s = √2 into the equation:

Area = 2s = 2(√2) = 2√2

Therefore, the area of the plank is 2√2 square meters.

However, none of the given options match this answer. So, there seems to be an error in the provided options, or the correct answer might be missing.