Let's start by gathering more information about these eight identical triangles.
First notice that we have 4 equal angles meeting at a single point.


So, each angle must be 90°


Now examine the red triangle below.


The red triangle is an isosceles triangle since all 4 sides of a square are equal.
The two equal angles must add to 90°
So, each angle must be 45°


Using similar logic, we can conclude that all of the eight triangles are 45-45-90 special right triangles.


Now let's examine what happens if we examine one particular 45-45-90 special right triangle, which has sides of length 1, 1 and √2


Use those lengths for all eight identical triangles in the 2 diagrams we get the following:


At this point, we can calculate the perimeters.
Perimeter of X = √2 + √2 + √2 + √2 = 4√2
Perimeter of Y = 1 + 1 + 1 + 1 + 1 + 1 = 6

So, perimeter of X : perimeter of Y = 4√2 : 6
Divide both sides by 2 to get the equivalent ratio 2√2 : 3

Answer: C

ASIDE: Please note that I didn't have to use those specific lengths (1, 1 and √2) for each of the 45-45-90 triangles.
I could have used ANY measurements that could be found in a 45-45-90 triangle.
For example, I could have used 5, 5 and 5√2, and those measurements still would have yielded the same ratio (after some simplifying)

Cheers,
Brent