To find the distance around the perimeter of the garden, we first need to analyze the geometric properties given in the problem. The garden is shaped like a rhombus, which can be divided into four identical [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangles. We are informed that the shorter distance across the middle of the garden is 30 feet. This middle distance is the length of the shorter diagonal of the rhombus.
Considering the properties of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, we know the following relationships between its sides:
- The side opposite the [tex]\(30^\circ\)[/tex] angle is the shortest side.
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\( \sqrt{3} \)[/tex] times the shortest side.
- The side opposite the [tex]\(90^\circ\)[/tex] angle (the hypotenuse) is twice the shortest side.
Given that the shorter distance across the middle of the garden (which is the shorter diagonal of the rhombus) is 30 feet, we recognize that this distance is composed of two sides of length corresponding to the shorter sides of the [tex]\(30^\circ\)[/tex] angles in the [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangles.
Thus, each of these shorter sides is:
[tex]\[ \text{shorter leg} = \frac{30 \text{ feet}}{2} = 15 \text{ feet} \][/tex]
Since the hypotenuse of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle is twice the length of the shorter leg, the side of the rhombus (which is the same as the hypotenuse of one of the triangles) is:
[tex]\[ \text{side length} = 2 \times 15 \text{ feet} = 30 \text{ feet} \][/tex]
To find the perimeter of the rhombus, we need to add up the lengths of all four sides:
[tex]\[ \text{perimeter} = 4 \times \text{side length} = 4 \times 30 \text{ feet} = 120 \text{ feet} \][/tex]
Therefore, the distance around the perimeter of the garden is:
[tex]\[ \boxed{120 \text{ feet}} \][/tex]
