Each Leg Of A $45^{\circ}-45^{\circ}-90^{\circ}$ Triangle Measures 14 Cm. What Is The Length Of The Hypotenuse?

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Introduction

In a right-angled triangle, the side opposite the right angle is called the hypotenuse. A special type of right-angled triangle is the 45-45-90 triangle, where the two legs are equal in length and the hypotenuse is √2 times the length of each leg. In this article, we will explore how to find the length of the hypotenuse of a 45-45-90 triangle when each leg measures 14 cm.

Understanding the 45-45-90 Triangle

A 45-45-90 triangle is a right-angled triangle with two equal legs and a hypotenuse that is √2 times the length of each leg. The two legs are equal in length, and the hypotenuse is the side opposite the right angle. The ratio of the sides in a 45-45-90 triangle is 1:1:√2.

Calculating the Hypotenuse

To find the length of the hypotenuse of a 45-45-90 triangle, we can use the ratio of the sides. Since each leg measures 14 cm, we can multiply the length of each leg by √2 to find the length of the hypotenuse.

The Formula

The formula to find the length of the hypotenuse of a 45-45-90 triangle is:

Hypotenuse = leg Γ— √2

Substituting the Values

Substituting the values, we get:

Hypotenuse = 14 Γ— √2

Simplifying the Expression

To simplify the expression, we can multiply 14 by √2.

14 Γ— √2 = 14√2

Evaluating the Expression

To evaluate the expression, we can approximate the value of √2.

√2 β‰ˆ 1.414

Multiplying the Values

Multiplying 14 by 1.414, we get:

14 Γ— 1.414 β‰ˆ 19.796

Rounding the Answer

Rounding the answer to the nearest whole number, we get:

19.796 β‰ˆ 20

Conclusion

In conclusion, the length of the hypotenuse of a 45-45-90 triangle with each leg measuring 14 cm is approximately 20 cm.

Real-World Applications

The 45-45-90 triangle has many real-world applications, including:

  • Construction: In construction, 45-45-90 triangles are used to calculate the length of rafters and roof beams.
  • Engineering: In engineering, 45-45-90 triangles are used to calculate the length of pipes and tubes.
  • Design: In design, 45-45-90 triangles are used to create symmetrical and balanced compositions.

Tips and Tricks

Here are some tips and tricks to help you solve problems involving 45-45-90 triangles:

  • Use the ratio of the sides: The ratio of the sides in a 45-45-90 triangle is 1:1:√2. Use this ratio to find the length of the hypotenuse.
  • Multiply by √2: To find the length of the hypotenuse, multiply the length of each leg by √2.
  • Approximate the value of √2: If you need to approximate the value of √2, use 1.414 as an approximation.

Practice Problems

Here are some practice problems to help you practice solving problems involving 45-45-90 triangles:

  • Problem 1: Find the length of the hypotenuse of a 45-45-90 triangle with each leg measuring 10 cm.
  • Problem 2: Find the length of the hypotenuse of a 45-45-90 triangle with each leg measuring 15 cm.
  • Problem 3: Find the length of the hypotenuse of a 45-45-90 triangle with each leg measuring 20 cm.

Answer Key

Here are the answers to the practice problems:

  • Problem 1: The length of the hypotenuse is approximately 14.14 cm.
  • Problem 2: The length of the hypotenuse is approximately 21.21 cm.
  • Problem 3: The length of the hypotenuse is approximately 28.28 cm.

Conclusion

Q: What is a 45-45-90 triangle?

A: A 45-45-90 triangle is a right-angled triangle with two equal legs and a hypotenuse that is √2 times the length of each leg.

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

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

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

A: To find the length of the hypotenuse of a 45-45-90 triangle, multiply the length of each leg by √2.

Q: What is the formula to find the length of the hypotenuse of a 45-45-90 triangle?

A: The formula to find the length of the hypotenuse of a 45-45-90 triangle is:

Hypotenuse = leg Γ— √2

Q: Can I use a calculator to find the length of the hypotenuse of a 45-45-90 triangle?

A: Yes, you can use a calculator to find the length of the hypotenuse of a 45-45-90 triangle. Simply multiply the length of each leg by √2.

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

A: Some real-world applications of 45-45-90 triangles include:

  • Construction: In construction, 45-45-90 triangles are used to calculate the length of rafters and roof beams.
  • Engineering: In engineering, 45-45-90 triangles are used to calculate the length of pipes and tubes.
  • Design: In design, 45-45-90 triangles are used to create symmetrical and balanced compositions.

Q: How do I approximate the value of √2?

A: You can approximate the value of √2 by using 1.414 as an approximation.

Q: What are some tips and tricks for solving problems involving 45-45-90 triangles?

A: Some tips and tricks for solving problems involving 45-45-90 triangles include:

  • Use the ratio of the sides: The ratio of the sides in a 45-45-90 triangle is 1:1:√2. Use this ratio to find the length of the hypotenuse.
  • Multiply by √2: To find the length of the hypotenuse, multiply the length of each leg by √2.
  • Approximate the value of √2: If you need to approximate the value of √2, use 1.414 as an approximation.

Q: What are some practice problems to help me practice solving problems involving 45-45-90 triangles?

A: Here are some practice problems to help you practice solving problems involving 45-45-90 triangles:

  • Problem 1: Find the length of the hypotenuse of a 45-45-90 triangle with each leg measuring 10 cm.
  • Problem 2: Find the length of the hypotenuse of a 45-45-90 triangle with each leg measuring 15 cm.
  • Problem 3: Find the length of the hypotenuse of a 45-45-90 triangle with each leg measuring 20 cm.

Q: What are the answers to the practice problems?

A: Here are the answers to the practice problems:

  • Problem 1: The length of the hypotenuse is approximately 14.14 cm.
  • Problem 2: The length of the hypotenuse is approximately 21.21 cm.
  • Problem 3: The length of the hypotenuse is approximately 28.28 cm.

Conclusion

In conclusion, the 45-45-90 triangle is a special type of right-angled triangle with two equal legs and a hypotenuse that is √2 times the length of each leg. By using the ratio of the sides and multiplying by √2, you can solve problems involving 45-45-90 triangles. Practice problems are provided to help you practice solving problems involving 45-45-90 triangles.