Certainly! Let's solve the problem step-by-step.
### Understanding the Problem:
- We have a garden shaped like a rhombus.
- The rhombus is formed from 4 identical [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangles.
- The shorter distance across the middle of the garden is 30 feet.
### Step-by-Step Solution:
1. Identify the Given Information:
- The shorter distance across the garden is 30 feet.
2. Understanding the Triangles:
- Each [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle has side ratios of [tex]\(1 : \sqrt{3} : 2\)[/tex].
- In each triangle, the side opposite the [tex]\(30^\circ\)[/tex] angle is half the hypotenuse.
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3}\)[/tex] times the side opposite the [tex]\(30^\circ\)[/tex] angle.
- The hypotenuse is twice the side opposite the [tex]\(30^\circ\)[/tex] angle.
3. Relating the Given Length to the Triangle:
- Since the shorter distance across the middle of the garden (30 feet) corresponds to the sum of the sides opposite the [tex]\(30^\circ\)[/tex] angles of two triangles, we can determine the length of the side opposite the [tex]\(30^\circ\)[/tex] angle.
4. Calculating the Lengths:
- The shorter distance (30 feet) is equal to twice the length of the side opposite the [tex]\(30^\circ\)[/tex] angle.
- Hence, the side opposite the [tex]\(30^\circ\)[/tex] angle (short side) is [tex]\( \frac{30}{2} = 15\)[/tex] feet.
5. Find the Lengths of the Other Sides:
- The side opposite the [tex]\(60^\circ\)[/tex] angle (middle side) is:
[tex]\[
\text{Middle side} = 15 \times \sqrt{3} \approx 25.98 \text{ feet}
\][/tex]
- The hypotenuse (opposite the [tex]\(90^\circ\)[/tex] angle) is:
[tex]\[
\text{Hypotenuse} = 15 \times 2 = 30 \text{ feet}
\][/tex]
6. Calculate the Area of the Rhombus:
- The area of one [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle is:
[tex]\[
\text{Area of one triangle} = \frac{1}{2} \times \text{short side} \times \text{middle side} = \frac{1}{2} \times 15 \times 25.98 \approx 194.86 \text{ square feet}
\][/tex]
- Since the rhombus is made up of 4 identical triangles, the total area of the rhombus is:
[tex]\[
\text{Total area} = 4 \times 194.86 \approx 779.42 \text{ square feet}
\][/tex]
### Final Results:
- [tex]\( \text{Short side opposite } 30^\circ \text{ angle: } 15 \text{ feet} \)[/tex]
- [tex]\( \text{Middle side opposite } 60^\circ \text{ angle: } 25.98 \text{ feet} \)[/tex]
- [tex]\( \text{Hypotenuse opposite } 90^\circ \text{ angle: } 30 \text{ feet} \)[/tex]
- [tex]\( \text{Total area of the rhombus: } 779.42 \text{ square feet} \)[/tex]
Thus, the side lengths and the area of the rhombus garden are as calculated above.
