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Sagot :
To find the dimensions and distances in this garden, let's analyze the geometry step-by-step.
1. Understand the Rhombus Formation: The garden is a rhombus, and it is subdivided into four identical [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangles. The shorter distance across the middle of the garden (30 feet) serves as the combined length of the shorter sides of two such triangles.
2. Apply Properties of [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] Triangle:
- In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex].
- Specifically, the side opposite the [tex]\(30^\circ\)[/tex] angle is the shortest side, denoted as [tex]\(x\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3} \cdot x\)[/tex].
- The hypotenuse (opposite the [tex]\(90^\circ\)[/tex] angle) is [tex]\(2x\)[/tex].
3. Determine the Lengths:
- Since the shorter distance across the rhombus is 30 feet, this corresponds to twice the shortest side of the two [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangles along that distance.
- Hence, the shorter side [tex]\(x\)[/tex] is half of 30 feet: [tex]\( x = \frac{30}{2} = 15 \)[/tex] feet.
4. Calculate the Remaining Sides:
- The longer leg (opposite the [tex]\(60^\circ\)[/tex] angle) in one triangle is [tex]\(\sqrt{3} \cdot x = \sqrt{3} \cdot 15\)[/tex] feet, or [tex]\( 15\sqrt{3} \)[/tex] feet.
- The length of the hypotenuse (which is also the side of the rhombus) is [tex]\(2x = 2 \cdot 15 = 30\)[/tex] feet.
5. Find the Longer Distance Across the Garden:
- The longer distance across the middle of the garden spans the longer legs of the triangles opposite the [tex]\(60^\circ\)[/tex] angles.
- Thus, the longer distance corresponds to [tex]\(2 \times 15\sqrt{3}\)[/tex] feet:
[tex]\[ 2 \times 15\sqrt{3} = 30\sqrt{3} \text{ feet}. \][/tex]
Summary:
- The shortest side of each [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle is 15 feet.
- The longer leg (opposite the [tex]\(60^\circ\)[/tex] angle) of each triangle is [tex]\(15\sqrt{3}\)[/tex] feet.
- The longer distance across the middle of the garden is [tex]\(30\sqrt{3}\)[/tex] feet.
So, the required dimensions and distances in the garden are:
- Shortest side of the triangles: 15 feet
- Longer leg of the triangles: [tex]\(15\sqrt{3}\)[/tex] feet
- Longer distance across the middle of the garden: [tex]\(30\sqrt{3}\)[/tex] feet
1. Understand the Rhombus Formation: The garden is a rhombus, and it is subdivided into four identical [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangles. The shorter distance across the middle of the garden (30 feet) serves as the combined length of the shorter sides of two such triangles.
2. Apply Properties of [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] Triangle:
- In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex].
- Specifically, the side opposite the [tex]\(30^\circ\)[/tex] angle is the shortest side, denoted as [tex]\(x\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3} \cdot x\)[/tex].
- The hypotenuse (opposite the [tex]\(90^\circ\)[/tex] angle) is [tex]\(2x\)[/tex].
3. Determine the Lengths:
- Since the shorter distance across the rhombus is 30 feet, this corresponds to twice the shortest side of the two [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangles along that distance.
- Hence, the shorter side [tex]\(x\)[/tex] is half of 30 feet: [tex]\( x = \frac{30}{2} = 15 \)[/tex] feet.
4. Calculate the Remaining Sides:
- The longer leg (opposite the [tex]\(60^\circ\)[/tex] angle) in one triangle is [tex]\(\sqrt{3} \cdot x = \sqrt{3} \cdot 15\)[/tex] feet, or [tex]\( 15\sqrt{3} \)[/tex] feet.
- The length of the hypotenuse (which is also the side of the rhombus) is [tex]\(2x = 2 \cdot 15 = 30\)[/tex] feet.
5. Find the Longer Distance Across the Garden:
- The longer distance across the middle of the garden spans the longer legs of the triangles opposite the [tex]\(60^\circ\)[/tex] angles.
- Thus, the longer distance corresponds to [tex]\(2 \times 15\sqrt{3}\)[/tex] feet:
[tex]\[ 2 \times 15\sqrt{3} = 30\sqrt{3} \text{ feet}. \][/tex]
Summary:
- The shortest side of each [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle is 15 feet.
- The longer leg (opposite the [tex]\(60^\circ\)[/tex] angle) of each triangle is [tex]\(15\sqrt{3}\)[/tex] feet.
- The longer distance across the middle of the garden is [tex]\(30\sqrt{3}\)[/tex] feet.
So, the required dimensions and distances in the garden are:
- Shortest side of the triangles: 15 feet
- Longer leg of the triangles: [tex]\(15\sqrt{3}\)[/tex] feet
- Longer distance across the middle of the garden: [tex]\(30\sqrt{3}\)[/tex] feet
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