Let's solve the problem step-by-step:
1. Understanding the Geometry: We have a rhombus composed of four [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangles. The shorter distance across the middle of the rhombus (which is actually one of its diagonals) measures 30 feet. In a rhombus, the diagonals bisect each other at right angles.
2. Identifying Properties of the [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] Triangle: In a [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle, the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex]. This means the side opposite the [tex]\(30^\circ\)[/tex] angle is half the length of the hypotenuse, and the side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3}\)[/tex] times the length of the side opposite the [tex]\(30^\circ\)[/tex] angle.
3. Calculating the Half of the Shorter Diagonal: The shorter diagonal of the rhombus (30 feet) is bisected into two equal segments by the intersection of the diagonals. Therefore, each segment of the shorter diagonal is [tex]\( \frac{30}{2} = 15 \)[/tex] feet.
4. Relating Sides in the 30-60-90 Triangle: The segment of the shorter diagonal (15 feet) represents the side opposite the [tex]\(30^\circ\)[/tex] angle in one of the [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangles that make up the rhombus.
5. Calculating the Side Length of the Rhombus: To find the side length of the rhombus, we use the properties of the [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle. The side opposite the [tex]\(30^\circ\)[/tex] angle is half the hypotenuse (or the side of the rhombus in this case). So, if the segment opposite the [tex]\(30^\circ\)[/tex] angle is 15 feet, the full side length of the rhombus (the hypotenuse of the triangle) is [tex]\( 15 \times 2 = 30 \)[/tex] feet.
6. Calculating the Perimeter: The rhombus has four sides of equal length. Therefore, the perimeter of the rhombus is:
[tex]\[ 4 \times \text{side length} = 4 \times 30 = 120 \text{ feet} \][/tex]
Thus, the distance around the perimeter of the rhombus-shaped garden is [tex]\(120\)[/tex] feet.
The correct answer is [tex]\(\boxed{120 \text{ ft}}\)[/tex].
