To find the relevant dimensions of the rhombus-shaped garden, let's analyze given information in the context of [tex]$30^{\circ}-60^{\circ}-90^{\circ}$[/tex] triangles.
### Step-by-Step Solution:
1. Identify the relationship of side lengths in a 30°-60°-90° triangle:
- These triangles have side lengths in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex]. Let's denote the shorter leg (opposite the 30° angle) as [tex]\(a\)[/tex], the longer leg (opposite the 60° angle) as [tex]\(a\sqrt{3}\)[/tex], and the hypotenuse as [tex]\(2a\)[/tex].
2. Define the given shorter diagonal of the rhombus:
- The rhombus is composed of 4 such triangles. The shorter diagonal of the rhombus runs between the shorter legs of two opposing triangles. Hence, the shorter diagonal is the cumulative length of two shorter legs.
- Given shorter diagonal [tex]\( = 30 \, \text{feet} \)[/tex].
3. Calculate the shorter leg [tex]\(a\)[/tex]:
- The shorter diagonal is formed by two shorter legs. Thus, [tex]\(2a = 30\)[/tex].
- Solving for [tex]\(a\)[/tex], we get [tex]\(a = \frac{30}{2} = 15 \, \text{feet}\)[/tex].
4. Determine the side length of the rhombus:
- The side of the rhombus is the hypotenuse of one of the 30°-60°-90° triangles, which is [tex]\(2a\)[/tex].
- Thus, the side length [tex]\( = 2 \times 15 = 30 \, \text{feet} \)[/tex].
5. Calculate the longer diagonal of the rhombus:
- The longer diagonal runs across the entire length of two hypotenuses of two opposing triangles.
- The longer diagonal [tex]\( = 2 \times \text{hypotenuse} = 2 \times 30 = 60 \, \text{feet} \)[/tex].
### Result:
- The shorter leg (a) is [tex]\(15 \, \text{feet}\)[/tex].
- The side length of the rhombus is [tex]\(30 \, \text{feet}\)[/tex].
- The longer diagonal of the rhombus is [tex]\(60 \, \text{feet}\)[/tex].
