The figure above consists of two equilateral triangles, one of which is inside the other. If the unshaded region within the larger triangle makes up 8/9 of that triangle's area, what is the ratio of the perimeter of the larger triangle to the perimeter of the smaller triangle?

( The figure looks like a smaller triangle (shaded) inscribed in a bigger triangle ( unshaded)

1/9 of the larger triangle's area is unshaded. That means the linear (edge) dimension of the smaller shaded triangle is 1/3 of the edge dimension of the larger triangle. The perimeters of the two triangles will also be in a large:small = 3:1 ratio.

To find the ratio of the perimeter of the larger triangle to the perimeter of the smaller triangle, we first need to determine the relationship between the areas of the two triangles.

Since the smaller triangle is inscribed within the larger triangle and both triangles are equilateral, we can conclude that the ratio of their areas is equal to the square of the ratio of their side lengths.

Let's assume that the side length of the smaller triangle is 's' and the side length of the larger triangle is 'S'.

Now, we know that the area of an equilateral triangle can be calculated using the formula:
Area = (√3/4) * (side length)^2

So, the area of the smaller triangle is (√3/4) * s^2, and the area of the larger triangle is (√3/4) * S^2.

Given that the shaded region within the larger triangle makes up 8/9 of its area, we can write the following equation:
(8/9) * (√3/4) * S^2 = (√3/4) * (S^2 - s^2)

To simplify, we can cancel out (√3/4) from both sides of the equation:
(8/9) * S^2 = S^2 - s^2

Next, let's solve for s^2:
s^2 = S^2 - (8/9) * S^2

Now, we can factor out an S^2 from the right side of the equation:
s^2 = S^2 * (1 - 8/9)

Simplifying further:
s^2 = S^2 * (1/9)

Taking the square root of both sides of the equation:
s = S/3

So, we have found that the side length of the smaller triangle is one-third the side length of the larger triangle.

The ratio of the perimeters of the two triangles is equal to the ratio of their side lengths. Therefore, the ratio of the perimeter of the larger triangle to the perimeter of the smaller triangle is:
S / s = S / (S/3) = 3

Hence, the ratio of the perimeter of the larger triangle to the perimeter of the smaller triangle is 3:1.