The Hypotenuse Of A $45 {\circ}-45 {\circ}-90^{\circ}$ Triangle Measures $2 \sqrt{2}$ Units.What Is The Length Of One Leg Of The Triangle?A. 11 Units B. $ 11 2 11 \sqrt{2} 11 2 ​ [/tex] Units C. 22 Units D. $22

by ADMIN 219 views

Introduction

In the realm of geometry, triangles are a fundamental concept that has been studied extensively. Among the various types of triangles, the 45°-45°-90° triangle is a special case that has unique properties. In this article, we will delve into the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle, with a specific focus on the given scenario where the hypotenuse measures 2√2 units.

Understanding the 45°-45°-90° Triangle

A 45°-45°-90° triangle is a right-angled triangle with two equal acute angles, each measuring 45°. This unique configuration results in the triangle having specific side length ratios. The sides of a 45°-45°-90° triangle are in the ratio 1:1:√2, where the hypotenuse is √2 times the length of each leg.

The Relationship Between Legs and Hypotenuse

Given that the hypotenuse of the triangle measures 2√2 units, we can use the ratio of the sides to determine the length of one leg. Since the hypotenuse is √2 times the length of each leg, we can set up a proportion to find the length of one leg.

Let x be the length of one leg. Then, the hypotenuse is x√2. Since the hypotenuse measures 2√2 units, we can set up the equation:

x√2 = 2√2

To solve for x, we can divide both sides of the equation by √2:

x = 2√2 / √2

x = 2

Therefore, the length of one leg of the triangle is 2 units.

Conclusion

In conclusion, the length of one leg of a 45°-45°-90° triangle with a hypotenuse measuring 2√2 units is 2 units. This is a direct result of the unique side length ratios of the 45°-45°-90° triangle, where the hypotenuse is √2 times the length of each leg.

Frequently Asked Questions

  • What is the length of one leg of a 45°-45°-90° triangle with a hypotenuse measuring 2√2 units?
  • The length of one leg of a 45°-45°-90° triangle with a hypotenuse measuring 2√2 units is 2 units.
  • What is the ratio of the sides of a 45°-45°-90° triangle?
  • The ratio of the sides of a 45°-45°-90° triangle is 1:1:√2.

Final Thoughts

In this article, we explored the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle. By understanding the unique side length ratios of this type of triangle, we can easily determine the length of one leg given the length of the hypotenuse. This knowledge is essential in various mathematical applications, including geometry and trigonometry.

References

  • [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [2] "Trigonometry: A Unit Circle Approach" by Michael Corral
  • [3] "Mathematics for the Nonmathematician" by Morris Kline

Additional Resources

  • Khan Academy: Geometry
  • MIT OpenCourseWare: Geometry
  • Wolfram MathWorld: 45°-45°-90° Triangle

Introduction

In our previous article, we explored the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle. We also provided a step-by-step solution to find the length of one leg given the length of the hypotenuse. In this article, we will address some frequently asked questions related to the topic.

Q&A

Q: What is the length of one leg of a 45°-45°-90° triangle with a hypotenuse measuring 2√2 units?

A: The length of one leg of a 45°-45°-90° triangle with a hypotenuse measuring 2√2 units is 2 units.

Q: What is the ratio of the sides of a 45°-45°-90° triangle?

A: The ratio of the sides of a 45°-45°-90° triangle is 1:1:√2.

Q: How do I find the length of one leg of a 45°-45°-90° triangle given the length of the hypotenuse?

A: To find the length of one leg of a 45°-45°-90° triangle given the length of the hypotenuse, you can use the ratio of the sides. Let x be the length of one leg. Then, the hypotenuse is x√2. Since the hypotenuse measures 2√2 units, you can set up the equation:

x√2 = 2√2

To solve for x, you can divide both sides of the equation by √2:

x = 2√2 / √2

x = 2

Therefore, the length of one leg of the triangle is 2 units.

Q: What are some real-world applications of 45°-45°-90° triangles?

A: 45°-45°-90° triangles have numerous real-world applications, including:

  • Building design: 45°-45°-90° triangles are used in the design of buildings to create strong and stable structures.
  • Engineering: 45°-45°-90° triangles are used in engineering to create complex systems and mechanisms.
  • Art: 45°-45°-90° triangles are used in art to create geometric patterns and designs.

Q: Can I use the same method to find the length of one leg of a 45°-45°-90° triangle with a hypotenuse measuring 3√2 units?

A: Yes, you can use the same method to find the length of one leg of a 45°-45°-90° triangle with a hypotenuse measuring 3√2 units. Let x be the length of one leg. Then, the hypotenuse is x√2. Since the hypotenuse measures 3√2 units, you can set up the equation:

x√2 = 3√2

To solve for x, you can divide both sides of the equation by √2:

x = 3√2 / √2

x = 3

Therefore, the length of one leg of the triangle is 3 units.

Conclusion

In this article, we addressed some frequently asked questions related to the topic of the hypotenuse of a 45°-45°-90° triangle. We provided step-by-step solutions to find the length of one leg given the length of the hypotenuse and discussed some real-world applications of 45°-45°-90° triangles.

Frequently Asked Questions

  • What is the length of one leg of a 45°-45°-90° triangle with a hypotenuse measuring 2√2 units?
  • The length of one leg of a 45°-45°-90° triangle with a hypotenuse measuring 2√2 units is 2 units.
  • What is the ratio of the sides of a 45°-45°-90° triangle?
  • The ratio of the sides of a 45°-45°-90° triangle is 1:1:√2.
  • How do I find the length of one leg of a 45°-45°-90° triangle given the length of the hypotenuse?
  • To find the length of one leg of a 45°-45°-90° triangle given the length of the hypotenuse, you can use the ratio of the sides.

Final Thoughts

In this article, we provided a comprehensive Q&A section related to the topic of the hypotenuse of a 45°-45°-90° triangle. We hope that this article has been helpful in addressing some of the most frequently asked questions related to this topic.

References

  • [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [2] "Trigonometry: A Unit Circle Approach" by Michael Corral
  • [3] "Mathematics for the Nonmathematician" by Morris Kline

Additional Resources

  • Khan Academy: Geometry
  • MIT OpenCourseWare: Geometry
  • Wolfram MathWorld: 45°-45°-90° Triangle