Possible Ratios of the Longer Leg to the Hypotenuse

Oct 13, 2025 by talynt.org 10 views

Introduction

In geometry, a 30-60-90 triangle is a special right-angled triangle with angles measuring 30, 60, and 90 degrees. These triangles have unique properties that make them useful in various mathematical and real-world applications. One of the key properties of 30-60-90 triangles is the ratio of the lengths of their sides. In this article, we will explore the possible ratios of the length of the longer leg of a 30-60-90 triangle to the length of its hypotenuse.

Understanding 30-60-90 Triangles

A 30-60-90 triangle is a right-angled triangle with one angle measuring 30 degrees and the other angle measuring 60 degrees. The side opposite the 30-degree angle is called the shorter leg, and the side opposite the 60-degree angle is called the longer leg. The side opposite the 90-degree angle is called the hypotenuse.

Properties of 30-60-90 Triangles

One of the key properties of 30-60-90 triangles is the ratio of the lengths of their sides. The ratio of the lengths of the sides of a 30-60-90 triangle is always 1:√3:2. This means that the ratio of the length of the shorter leg to the length of the longer leg is 1:√3, and the ratio of the length of the longer leg to the length of the hypotenuse is 1:2.

Possible Ratios of the Longer Leg to the Hypotenuse

Given the properties of 30-60-90 triangles, we can determine the possible ratios of the length of the longer leg to the length of the hypotenuse. Since the ratio of the length of the longer leg to the length of the hypotenuse is 1:2, we can express this ratio as a fraction: 1/2.

Checking the Options

Now, let's check the options provided to see which ones match the possible ratios of the length of the longer leg to the length of the hypotenuse.

Option A: 2: 2√2

This option does not match the possible ratio of the length of the longer leg to the length of the hypotenuse. The ratio 2: 2√2 is not equal to 1:2.

Option B: 1: √2

This option does not match the possible ratio of the length of the longer leg to the length of the hypotenuse. The ratio 1: √2 is not equal to 1:2.

Option C: 1: 2

This option matches the possible ratio of the length of the longer leg to the length of the hypotenuse. The ratio 1: 2 is equal to 1:2.

Conclusion

In conclusion, the possible ratio of the length of the longer leg of a 30-60-90 triangle to the length of its hypotenuse is 1:2. This ratio is a fundamental property of 30-60-90 triangles and is useful in various mathematical and real-world applications.

Final Answer

The final answer is:

Option C: 1: 2
Frequently Asked Questions About 30-60-90 Triangles =====================================================

Introduction

In our previous article, we explored the properties of 30-60-90 triangles and determined the possible ratio of the length of the longer leg to the length of the hypotenuse. In this article, we will answer some frequently asked questions about 30-60-90 triangles.

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

A 30-60-90 triangle is a special right-angled triangle with angles measuring 30, 60, and 90 degrees.

Q: What are the properties of a 30-60-90 triangle?

One of the key properties of a 30-60-90 triangle is the ratio of the lengths of its sides. The ratio of the lengths of the sides of a 30-60-90 triangle is always 1:√3:2.

Q: What is the ratio of the length of the longer leg to the length of the hypotenuse in a 30-60-90 triangle?

The ratio of the length of the longer leg to the length of the hypotenuse in a 30-60-90 triangle is 1:2.

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

To identify a 30-60-90 triangle, look for the following characteristics:

One angle measures 30 degrees
One angle measures 60 degrees
One angle measures 90 degrees
The ratio of the lengths of the sides is 1:√3:2

Q: What are the applications of 30-60-90 triangles?

30-60-90 triangles have many applications in mathematics and real-world situations, including:

Building design and construction
Engineering and architecture
Geometry and trigonometry
Physics and engineering

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

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

Q: How do I use a 30-60-90 triangle to solve problems?

To use a 30-60-90 triangle to solve problems, follow these steps:

Identify the problem and determine the type of triangle required.
Draw the triangle and label the angles and sides.
Use the properties of the triangle to solve the problem.

Q: What are some common mistakes to avoid when working with 30-60-90 triangles?

Some common mistakes to avoid when working with 30-60-90 triangles include:

Confusing the ratio of the lengths of the sides
Failing to identify the type of triangle required
Not using the properties of the triangle to solve the problem

Conclusion

In conclusion, 30-60-90 triangles are a fundamental concept in mathematics and have many applications in real-world situations. By understanding the properties and characteristics of 30-60-90 triangles, you can use them to solve problems in mathematics and other areas.

Final Answer

The final answer is:

A 30-60-90 triangle is a special right-angled triangle with angles measuring 30, 60, and 90 degrees.
The ratio of the lengths of the sides of a 30-60-90 triangle is always 1:√3:2.
The ratio of the length of the longer leg to the length of the hypotenuse in a 30-60-90 triangle is 1:2.