Figure X and Y above show how eight identical triangular pieces…
We are given that square X and rectangle Y were created with 8 identical triangles. Let’s sketch the figures below.
We use the fact that the two diagonals of a square always form right angles where they intersect. Thus, we see that square X is composed of 4 identical 45-45-90 right triangles and rectangle Y is also composed of 4 of these 45-45-90 right triangles. Since 45-45-90 right triangles have a side ratio of x: x: x√2, we can label the sides of our triangles in the two figures. Notice that the sides of the square are the hypotenuses of the triangles and the sides of the rectangle are the legs of the triangles.
Letting n be the length of the side of the 45-45-90 triangle, we label the diagram as shown:
We have what we need to determine the perimeter of square X and rectangle Y.
Perimeter of X = n√2 + n√2 + n√2 + n√2 = 4n√2
Perimeter of Y = n + n + n + n + n + n = 6n
Finally we must determine the ratio of the perimeter of square X to rectangle Y.
(Perimeter of X)/(Perimeter of Y) = (4n√2)/(6n) = 2√2/3, or, equivalently, 2√2 : 3