A Certain Triangle Has A 30° Angle And A 60° Angle. Which Must Be A True Statement About The Triangle?A. The Longest Side Is Twice As Long As The Second-longest Side.B. Two Sides Of The Triangle Have The Same Length.C. The Longest Side Is 5 Times As

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Introduction

In geometry, triangles are fundamental shapes that have been studied for centuries. One of the most interesting properties of triangles is the relationship between their angles and sides. In this article, we will explore a specific type of triangle that has a 30° angle and a 60° angle. We will examine three possible statements about this triangle and determine which one must be true.

Understanding the Triangle

A triangle with a 30° angle and a 60° angle is a special type of triangle known as a 30-60-90 triangle. This type of triangle has a unique property where the ratio of the sides is always 1:√3:2. The side opposite the 30° angle is the shortest side, the side opposite the 60° angle is the middle side, and the side opposite the 90° angle is the longest side.

Statement A: The Longest Side is Twice as Long as the Second-Longest Side

Let's examine statement A, which claims that the longest side is twice as long as the second-longest side. In a 30-60-90 triangle, the ratio of the sides is 1:√3:2. This means that the side opposite the 30° angle is 1 unit, the side opposite the 60° angle is √3 units, and the side opposite the 90° angle is 2 units. Since the side opposite the 90° angle is twice as long as the side opposite the 60° angle, statement A is actually true.

Statement B: Two Sides of the Triangle Have the Same Length

Statement B claims that two sides of the triangle have the same length. However, in a 30-60-90 triangle, the sides have a specific ratio of 1:√3:2. This means that the side opposite the 30° angle is not the same length as the side opposite the 60° angle, and the side opposite the 60° angle is not the same length as the side opposite the 90° angle. Therefore, statement B is false.

Statement C: The Longest Side is 5 Times as Long as the Second-Longest Side

Statement C claims that the longest side is 5 times as long as the second-longest side. However, in a 30-60-90 triangle, the ratio of the sides is 1:√3:2. This means that the side opposite the 90° angle is twice as long as the side opposite the 60° angle, not 5 times as long. Therefore, statement C is false.

Conclusion

In conclusion, statement A is the only true statement about the triangle. The longest side is indeed twice as long as the second-longest side, as determined by the ratio of the sides in a 30-60-90 triangle. This is a fundamental property of this type of triangle, and it is essential to understand this property in order to solve problems involving 30-60-90 triangles.

Applications of 30-60-90 Triangles

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 rafters.
  • Engineering: 30-60-90 triangles are used in engineering to calculate the stress and strain on materials.
  • Surveying: 30-60-90 triangles are used in surveying to calculate the distance between two points.
  • Geometry: 30-60-90 triangles are used in geometry to calculate the area and perimeter of triangles.

Examples of 30-60-90 Triangles

Here are some examples of 30-60-90 triangles:

  • A 30-60-90 triangle with a hypotenuse of 10 units: In this triangle, the side opposite the 30° angle is 5 units, the side opposite the 60° angle is 5√3 units, and the side opposite the 90° angle is 10 units.
  • A 30-60-90 triangle with a hypotenuse of 15 units: In this triangle, the side opposite the 30° angle is 7.5 units, the side opposite the 60° angle is 12.99 units, and the side opposite the 90° angle is 15 units.

Conclusion

In conclusion, 30-60-90 triangles are a fundamental concept in geometry, and understanding their properties is essential for solving problems involving these triangles. Statement A is the only true statement about the triangle, and it is a fundamental property of 30-60-90 triangles.

Introduction

In our previous article, we explored the properties of a triangle with a 30° angle and a 60° angle, known as a 30-60-90 triangle. We examined three possible statements about this triangle and determined which one must be true. In this article, we will answer some frequently asked questions about 30-60-90 triangles.

Q: What is a 30-60-90 triangle?

A: A 30-60-90 triangle is a type of triangle that has a 30° angle and a 60° angle. The side opposite the 30° angle is the shortest side, the side opposite the 60° angle is the middle side, and the side opposite the 90° angle is the longest side.

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

A: The ratio of the sides in a 30-60-90 triangle is 1:√3:2. This means that the side opposite the 30° angle is 1 unit, the side opposite the 60° angle is √3 units, and the side opposite the 90° angle is 2 units.

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

A: To calculate the length of the sides in a 30-60-90 triangle, you can use the ratio of the sides. For example, if the hypotenuse (the side opposite the 90° angle) is 10 units, then the side opposite the 30° angle is 5 units, and the side opposite the 60° angle is 5√3 units.

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 rafters.
  • Engineering: 30-60-90 triangles are used in engineering to calculate the stress and strain on materials.
  • Surveying: 30-60-90 triangles are used in surveying to calculate the distance between two points.
  • 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 calculate the area of a triangle?

A: Yes, you can use a 30-60-90 triangle to calculate the area of a triangle. The area of a triangle is equal to half the product of the base and the height. In a 30-60-90 triangle, the base is the side opposite the 30° angle, and the height is the side opposite the 60° angle.

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

A: To identify a 30-60-90 triangle, look for a triangle with a 30° angle and a 60° angle. The side opposite the 30° angle is the shortest side, the side opposite the 60° angle is the middle side, and the side opposite the 90° angle is the longest side.

Q: Can I use a 30-60-90 triangle to calculate the perimeter of a triangle?

A: Yes, you can use a 30-60-90 triangle to calculate the perimeter of a triangle. The perimeter of a triangle is equal to the sum of the lengths of the sides. In a 30-60-90 triangle, the perimeter is equal to the sum of the lengths of the three sides.

Conclusion

In conclusion, 30-60-90 triangles are a fundamental concept in geometry, and understanding their properties is essential for solving problems involving these triangles. We hope that this Q&A article has helped to answer some of your questions about 30-60-90 triangles.