To determine the perimeter of the garden, we can use the properties of the [tex]$30^{\circ}-60^{\circ}-90^{\circ}$[/tex] triangles that form the rhombus. Let’s break the problem down step by step:
1. Understand the Geometry of the Rhombus:
- The rhombus is composed of four [tex]$30^{\circ}-60^{\circ}-90^{\circ}$[/tex] triangles.
- The shorter distance across the middle of the rhombus, which is 30 feet, connects two opposite vertices and is the sum of the short legs of two of these triangles.
2. Properties of the [tex]$30^{\circ}-60^{\circ}-90^{\circ}$[/tex] Triangles:
- In a [tex]$30^{\circ}-60^{\circ}-90^{\circ}$[/tex] triangle, the sides have a specific ratio:
- The length of the short leg (opposite the [tex]$30^{\circ}$[/tex] angle) is half the length of the hypotenuse.
- The length of the long leg (opposite the [tex]$60^{\circ}$[/tex] angle) is [tex]$\sqrt{3}$[/tex] times the short leg.
3. Calculate the Short Leg:
- Given the shorter distance across the middle of 30 feet, this distance is made up of the short legs of two of the triangles.
- So, the length of the short leg of each triangle is:
[tex]\[
\text{short leg} = \frac{30}{2} = 15 \text{ feet}
\][/tex]
4. Calculate the Hypotenuse:
- In a [tex]$30^{\circ}-60^{\circ}-90^{\circ}$[/tex] triangle, the hypotenuse is twice the length of the short leg.
- Therefore, the hypotenuse (which serves as the side of the rhombus) is:
[tex]\[
\text{hypotenuse} = 2 \times 15 = 30 \text{ feet}
\][/tex]
5. Calculate the Perimeter of the Rhombus:
- The rhombus has four sides, each equal to the hypotenuse of one of the triangles.
- Thus, the perimeter of the rhombus is:
[tex]\[
\text{perimeter} = 4 \times 30 = 120 \text{ feet}
\][/tex]
So, the distance around the perimeter of the garden is:
[tex]\[
\boxed{120 \text{ feet}}
\][/tex]
