Given Info: We are given two figures, one rectangle and another square which are formed by identical triangular pieces. We need to find the ratio of perimeters of these 2 figures.

Interpreting the Problem: In order to find the ratio of the perimeter of these 2 figures, we have to first workout the sides of the identical triangles which form the square and triangle.

Solution: Finding the sides of the identical triangle in terms of sides of the square. Let us assume the side of the square to be a. Now the diagonal of the square will be \(a\sqrt{2}\). Now since the diagonals of the square bisect each other, side of the identical triangle will be \(a\sqrt{2}/2\).

The other side of the triangle will be the side of the square i.e. a.

The sides of identical triangles is shown in the figure.

Attachment:

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Now Calculating the perimeter of Figure X.

Perimeter of the square will be 4a

Calculating the perimeter of Figure Y.

Perimeter of rectangle will be \(2a\sqrt{2}/2\) + \(4a\sqrt{2}/2\) = \(3a\sqrt{2}\)

Ratio of perimeter of both figures

\(Perimeter X/Perimeter Y\) = 4a:\(3a\sqrt{2}\) =\(2\sqrt{2}:3\)

Hence, option C is correct.