A Garden Is Designed In The Shape Of A Rhombus Formed From 4 Identical \[$30^{\circ}-60^{\circ}-90^{\circ}\$\] Triangles. The Shorter Distance Across The Middle Of The Garden Measures 30 Feet.What Is The Distance Around The Perimeter Of The
Introduction
A rhombus is a type of polygon that has four equal sides and opposite angles that are equal. In this problem, we are given a garden designed in the shape of a rhombus formed from 4 identical 30°-60°-90° triangles. The shorter distance across the middle of the garden measures 30 feet. Our goal is to find the distance around the perimeter of the garden.
Understanding the Rhombus
A rhombus can be divided into two congruent triangles by drawing a diagonal from one vertex to the opposite vertex. In this case, we have a rhombus formed from 4 identical 30°-60°-90° triangles. This means that each triangle has angles of 30°, 60°, and 90°, and the sides opposite these angles are in the ratio 1:√3:2.
Calculating the Side Length
To find the side length of the rhombus, we need to find the length of one of the sides of the 30°-60°-90° triangle. Since the shorter distance across the middle of the garden measures 30 feet, this distance is equal to the length of the base of two of the triangles. Therefore, the length of the base of one triangle is half of 30 feet, which is 15 feet.
Using the Properties of 30°-60°-90° Triangles
In a 30°-60°-90° triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is √3 times the length of the side opposite the 30° angle. Since the length of the base of one triangle is 15 feet, the length of the side opposite the 60° angle is 15√3 feet.
Finding the Side Length of the Rhombus
Since the rhombus is formed from 4 identical 30°-60°-90° triangles, the side length of the rhombus is equal to the length of the side opposite the 60° angle, which is 15√3 feet.
Calculating the Perimeter
The perimeter of a rhombus is equal to the sum of the lengths of its four sides. Since the side length of the rhombus is 15√3 feet, the perimeter is 4 times this length, which is 4(15√3) feet.
Simplifying the Perimeter
To simplify the perimeter, we can multiply the numbers and keep the variable in the same position. Therefore, the perimeter is 60√3 feet.
Conclusion
In this problem, we were given a garden designed in the shape of a rhombus formed from 4 identical 30°-60°-90° triangles. The shorter distance across the middle of the garden measures 30 feet. We used the properties of 30°-60°-90° triangles to find the side length of the rhombus, and then calculated the perimeter of the garden. The perimeter of the garden is 60√3 feet.
Calculating the Perimeter in Decimal Form
To find the perimeter in decimal form, we can use a calculator to evaluate the expression 60√3. The value of √3 is approximately 1.732, so the perimeter is approximately 60(1.732) = 103.92 feet.
Real-World Applications
This problem has real-world applications in architecture, engineering, and design. For example, a builder may need to calculate the perimeter of a rhombus-shaped garden to determine the amount of materials needed for construction. An engineer may need to calculate the perimeter of a rhombus-shaped bridge to determine its structural integrity.
Conclusion
In conclusion, this problem demonstrates the importance of understanding the properties of 30°-60°-90° triangles and how to apply them to real-world problems. By using the properties of these triangles, we can calculate the side length and perimeter of a rhombus-shaped garden, and apply this knowledge to a variety of real-world applications.
Final Answer
Introduction
In our previous article, we explored the problem of calculating the perimeter of a rhombus-shaped garden formed from 4 identical 30°-60°-90° triangles. The shorter distance across the middle of the garden measures 30 feet. We used the properties of 30°-60°-90° triangles to find the side length of the rhombus, and then calculated the perimeter of the garden. In this article, we will answer some frequently asked questions related to this problem.
Q: What is a rhombus?
A: A rhombus is a type of polygon that has four equal sides and opposite angles that are equal. It can be divided into two congruent triangles by drawing a diagonal from one vertex to the opposite vertex.
Q: What is a 30°-60°-90° triangle?
A: A 30°-60°-90° triangle is a type of right triangle that has angles of 30°, 60°, and 90°. The sides opposite these angles are in the ratio 1:√3:2.
Q: How do I find the side length of a rhombus formed from 30°-60°-90° triangles?
A: To find the side length of a rhombus formed from 30°-60°-90° triangles, you need to find the length of one of the sides of the 30°-60°-90° triangle. Since the shorter distance across the middle of the garden measures 30 feet, this distance is equal to the length of the base of two of the triangles. Therefore, the length of the base of one triangle is half of 30 feet, which is 15 feet.
Q: How do I calculate the perimeter of a rhombus?
A: The perimeter of a rhombus is equal to the sum of the lengths of its four sides. Since the side length of the rhombus is 15√3 feet, the perimeter is 4 times this length, which is 4(15√3) feet.
Q: Can I simplify the perimeter?
A: Yes, you can simplify the perimeter by multiplying the numbers and keeping the variable in the same position. Therefore, the perimeter is 60√3 feet.
Q: How do I calculate the perimeter in decimal form?
A: To find the perimeter in decimal form, you can use a calculator to evaluate the expression 60√3. The value of √3 is approximately 1.732, so the perimeter is approximately 60(1.732) = 103.92 feet.
Q: What are some real-world applications of this problem?
A: This problem has real-world applications in architecture, engineering, and design. For example, a builder may need to calculate the perimeter of a rhombus-shaped garden to determine the amount of materials needed for construction. An engineer may need to calculate the perimeter of a rhombus-shaped bridge to determine its structural integrity.
Q: Can I use this method to calculate the perimeter of other shapes?
A: Yes, you can use this method to calculate the perimeter of other shapes that are formed from 30°-60°-90° triangles. However, you will need to adjust the formula to account for the specific shape and its dimensions.
Conclusion
In this article, we answered some frequently asked questions related to the problem of calculating the perimeter of a rhombus-shaped garden formed from 4 identical 30°-60°-90° triangles. We hope that this article has provided you with a better understanding of the problem and its solutions.
Final Answer
The final answer is: