Recall That In A $30^{\circ}-60^{\circ}-90^{\circ}$ Triangle, If The Shortest Leg Measures $x$ Units, Then The Longer Leg Measures $x \sqrt{3}$ Units And The Hypotenuse Measures $2 X$ Units.Choose The Correct

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A 30°-60°-90° triangle is a special type of right-angled triangle that has specific properties and ratios between its sides. In this article, we will explore the properties of a 30°-60°-90° triangle and how to use them to solve problems.

Recall the Properties of a 30°-60°-90° Triangle

A 30°-60°-90° triangle is a right-angled triangle with one angle measuring 30°, one angle measuring 60°, and the third angle measuring 90°. The sides of a 30°-60°-90° triangle are in a specific ratio, which is 1:√3:2. This means that if the shortest leg measures x units, then the longer leg measures x√3 units and the hypotenuse measures 2x units.

The Shortest Leg

The shortest leg of a 30°-60°-90° triangle is the side opposite the 30° angle. This side is also known as the base of the triangle. The length of the shortest leg is given as x units.

The Longer Leg

The longer leg of a 30°-60°-90° triangle is the side opposite the 60° angle. This side is also known as the height of the triangle. The length of the longer leg is given as x√3 units.

The Hypotenuse

The hypotenuse of a 30°-60°-90° triangle is the side opposite the 90° angle. This side is also known as the diagonal of the triangle. The length of the hypotenuse is given as 2x units.

Example Problems

Problem 1

In a 30°-60°-90° triangle, the shortest leg measures 6 units. What is the length of the longer leg?

Solution

Since the shortest leg measures 6 units, we can use the ratio of the sides to find the length of the longer leg. The ratio of the sides is 1:√3:2, so the length of the longer leg is 6√3 units.

Problem 2

In a 30°-60°-90° triangle, the hypotenuse measures 12 units. What is the length of the shortest leg?

Solution

Since the hypotenuse measures 12 units, we can use the ratio of the sides to find the length of the shortest leg. The ratio of the sides is 1:√3:2, so the length of the shortest leg is 6 units.

Real-World Applications

30°-60°-90° triangles have many real-world applications, including:

  • Construction: 30°-60°-90° triangles are used in construction to calculate the height of buildings and the length of roofs.
  • Engineering: 30°-60°-90° triangles are used in engineering to calculate the stress and strain on materials.
  • Geometry: 30°-60°-90° triangles are used in geometry to calculate the area and perimeter of triangles.

Conclusion

In conclusion, 30°-60°-90° triangles are a special type of right-angled triangle that has specific properties and ratios between its sides. By understanding the properties of a 30°-60°-90° triangle, we can solve problems and apply the concepts to real-world situations.

Key Takeaways

  • A 30°-60°-90° triangle is a right-angled triangle with one angle measuring 30°, one angle measuring 60°, and the third angle measuring 90°.
  • The sides of a 30°-60°-90° triangle are in a specific ratio, which is 1:√3:2.
  • The shortest leg measures x units, the longer leg measures x√3 units, and the hypotenuse measures 2x units.
  • 30°-60°-90° triangles have many real-world applications, including construction, engineering, and geometry.

Frequently Asked Questions

Q: What is a 30°-60°-90° triangle?

A: A 30°-60°-90° triangle is a right-angled triangle with one angle measuring 30°, one angle measuring 60°, and the third angle measuring 90°.

Q: What is the ratio of the sides of a 30°-60°-90° triangle?

A: The ratio of the sides of a 30°-60°-90° triangle is 1:√3:2.

Q: What is the length of the shortest leg of a 30°-60°-90° triangle?

A: The length of the shortest leg of a 30°-60°-90° triangle is x units.

Q: What is the length of the longer leg of a 30°-60°-90° triangle?

A: The length of the longer leg of a 30°-60°-90° triangle is x√3 units.

Q: What is the length of the hypotenuse of a 30°-60°-90° triangle?

A 30°-60°-90° triangle is a special type of right-angled triangle that has specific properties and ratios between its sides. In this article, we will answer some of the most frequently asked questions about 30°-60°-90° triangles.

Q: What is a 30°-60°-90° triangle?

A: A 30°-60°-90° triangle is a right-angled triangle with one angle measuring 30°, one angle measuring 60°, and the third angle measuring 90°.

Q: What is the ratio of the sides of a 30°-60°-90° triangle?

A: The ratio of the sides of a 30°-60°-90° triangle is 1:√3:2. This means that if the shortest leg measures x units, then the longer leg measures x√3 units and the hypotenuse measures 2x units.

Q: What is the length of the shortest leg of a 30°-60°-90° triangle?

A: The length of the shortest leg of a 30°-60°-90° triangle is x units.

Q: What is the length of the longer leg of a 30°-60°-90° triangle?

A: The length of the longer leg of a 30°-60°-90° triangle is x√3 units.

Q: What is the length of the hypotenuse of a 30°-60°-90° triangle?

A: The length of the hypotenuse of a 30°-60°-90° triangle is 2x units.

Q: How do I find the length of the sides of a 30°-60°-90° triangle?

A: To find the length of the sides of a 30°-60°-90° triangle, you can use the ratio of the sides. If you know the length of one side, you can use the ratio to find the length of the other sides.

Q: What are some real-world applications of 30°-60°-90° triangles?

A: 30°-60°-90° triangles have many real-world applications, including:

  • Construction: 30°-60°-90° triangles are used in construction to calculate the height of buildings and the length of roofs.
  • Engineering: 30°-60°-90° triangles are used in engineering to calculate the stress and strain on materials.
  • Geometry: 30°-60°-90° triangles are used in geometry to calculate the area and perimeter of triangles.

Q: Can I use a 30°-60°-90° triangle to solve problems in other areas of mathematics?

A: Yes, you can use a 30°-60°-90° triangle to solve problems in other areas of mathematics, such as trigonometry and algebra.

Q: How do I identify a 30°-60°-90° triangle?

A: To identify a 30°-60°-90° triangle, you can look for the following characteristics:

  • The triangle has one angle measuring 30°, one angle measuring 60°, and the third angle measuring 90°.
  • The sides of the triangle are in a ratio of 1:√3:2.
  • The shortest leg is opposite the 30° angle, the longer leg is opposite the 60° angle, and the hypotenuse is opposite the 90° angle.

Q: Can I use a 30°-60°-90° triangle to solve problems in real-world situations?

A: Yes, you can use a 30°-60°-90° triangle to solve problems in real-world situations, such as calculating the height of a building or the length of a roof.

Conclusion

In conclusion, 30°-60°-90° triangles are a special type of right-angled triangle that has specific properties and ratios between its sides. By understanding the properties of a 30°-60°-90° triangle, you can solve problems and apply the concepts to real-world situations.

Key Takeaways

  • A 30°-60°-90° triangle is a right-angled triangle with one angle measuring 30°, one angle measuring 60°, and the third angle measuring 90°.
  • The ratio of the sides of a 30°-60°-90° triangle is 1:√3:2.
  • The shortest leg measures x units, the longer leg measures x√3 units, and the hypotenuse measures 2x units.
  • 30°-60°-90° triangles have many real-world applications, including construction, engineering, and geometry.

Additional Resources

If you want to learn more about 30°-60°-90° triangles, here are some additional resources:

  • Online tutorials: There are many online tutorials and videos that can help you learn about 30°-60°-90° triangles.
  • Math textbooks: There are many math textbooks that cover the topic of 30°-60°-90° triangles.
  • Math websites: There are many math websites that provide information and resources on 30°-60°-90° triangles.

Final Thoughts

In conclusion, 30°-60°-90° triangles are a special type of right-angled triangle that has specific properties and ratios between its sides. By understanding the properties of a 30°-60°-90° triangle, you can solve problems and apply the concepts to real-world situations.