To find the longer distance across the middle of the rhombus, we'll follow these steps:
1. Identify the Rhombus Structure:
The rhombus is composed of four identical [tex]\( 30^{\circ}-60^{\circ}-90^{\circ} \)[/tex] triangles.
2. Understand the Key Properties of a [tex]\( 30^{\circ}-60^{\circ}-90^{\circ} \)[/tex] Triangle:
In a [tex]\( 30^{\circ}-60^{\circ}-90^{\circ} \)[/tex] triangle, the sides are in a fixed ratio:
- The side opposite the [tex]\( 30^{\circ} \)[/tex] angle (shorter leg) is [tex]\( x \)[/tex].
- The side opposite the [tex]\( 60^{\circ} \)[/tex] angle (longer leg) is [tex]\( x\sqrt{3} \)[/tex].
- The hypotenuse is [tex]\( 2x \)[/tex].
3. Given Information:
The shorter distance across the middle of the rhombus (which corresponds to the shorter leg, [tex]\( x \)[/tex], of one of the triangles) is 30 feet.
4. Determine the Values for the Sides:
Since the shorter leg of the [tex]\( 30^{\circ}-60^{\circ}-90^{\circ} \)[/tex] triangle is 30 feet:
- Shorter leg [tex]\( (x) = 30 \)[/tex] feet.
5. Calculate the Longer Leg:
Using the properties of the [tex]\( 30^{\circ}-60^{\circ}-90^{\circ} \)[/tex] triangle, the longer leg [tex]\( x\sqrt{3} \)[/tex] can be found as:
[tex]\[
\text{Longer leg} = 30 \times \sqrt{3} \approx 51.9615 \text{ feet}
\][/tex]
6. Relate This to the Rhombus Structure:
- Each triangle’s longer leg is half of the longer distance across the rhombus.
- Since this longer leg forms half of the longer diagonal of the rhombus, the full longer distance across the rhombus is twice the longer leg.
7. Calculate the Longer Distance Across the Rhombus:
[tex]\[
\text{Longer distance} = 2 \times (\text{Longer leg}) = 2 \times 51.9615 \approx 103.9230 \text{ feet}
\][/tex]
Finally, the longer distance across the middle of the rhombus is approximately 103.9230 feet.
