Each Leg Of A $45 {\circ}-45 {\circ}-90^{\circ}$ Triangle Measures 12 Cm.What Is The Length Of The Hypotenuse?A. 6 Cm B. $5 \sqrt{2}$ Cm C. 12 Cm D. $ 12 2 12 \sqrt{2} 12 2 ​ [/tex] Cm

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Introduction

In this article, we will explore the properties of a 45-45-90 triangle and use this knowledge to find the length of the hypotenuse. A 45-45-90 triangle is a special right triangle with two 45-degree angles and one 90-degree angle. This type of triangle has several unique properties that make it easy to solve. We will use the given information that each leg of the triangle measures 12 cm to find the length of the hypotenuse.

Understanding 45-45-90 Triangles

A 45-45-90 triangle is a right triangle with two 45-degree angles and one 90-degree angle. The two 45-degree angles are equal, and the hypotenuse is the side opposite the 90-degree angle. In a 45-45-90 triangle, the two legs are equal in length, and the hypotenuse is √2 times the length of each leg.

Properties of 45-45-90 Triangles

The properties of a 45-45-90 triangle are as follows:

  • The two legs are equal in length.
  • The hypotenuse is √2 times the length of each leg.
  • The two 45-degree angles are equal.
  • The 90-degree angle is opposite the hypotenuse.

Finding the Length of the Hypotenuse

To find the length of the hypotenuse, we can use the properties of a 45-45-90 triangle. Since each leg measures 12 cm, we can multiply the length of each leg by √2 to find the length of the hypotenuse.

Calculating the Length of the Hypotenuse

The length of the hypotenuse can be calculated as follows:

Length of hypotenuse = √2 * length of each leg = √2 * 12 cm = 12√2 cm

Conclusion

In this article, we explored the properties of a 45-45-90 triangle and used this knowledge to find the length of the hypotenuse. We found that the length of the hypotenuse is 12√2 cm. This is a common type of problem in geometry and trigonometry, and understanding the properties of 45-45-90 triangles can help you solve similar problems.

Example Problems

Here are a few example problems that you can try to practice your skills:

  • Find the length of the hypotenuse of a 45-45-90 triangle with legs measuring 15 cm.
  • Find the length of each leg of a 45-45-90 triangle with a hypotenuse measuring 21 cm.
  • Find the length of the hypotenuse of a 45-45-90 triangle with legs measuring 20 cm.

Solutions to Example Problems

Here are the solutions to the example problems:

  • Find the length of the hypotenuse of a 45-45-90 triangle with legs measuring 15 cm.

Length of hypotenuse = √2 * length of each leg = √2 * 15 cm = 15√2 cm

  • Find the length of each leg of a 45-45-90 triangle with a hypotenuse measuring 21 cm.

Length of each leg = length of hypotenuse / √2 = 21 cm / √2 = 21√2 / 2 cm

  • Find the length of the hypotenuse of a 45-45-90 triangle with legs measuring 20 cm.

Length of hypotenuse = √2 * length of each leg = √2 * 20 cm = 20√2 cm

Practice Problems

Here are a few practice problems that you can try to practice your skills:

  • Find the length of the hypotenuse of a 45-45-90 triangle with legs measuring 18 cm.
  • Find the length of each leg of a 45-45-90 triangle with a hypotenuse measuring 24 cm.
  • Find the length of the hypotenuse of a 45-45-90 triangle with legs measuring 22 cm.

Solutions to Practice Problems

Here are the solutions to the practice problems:

  • Find the length of the hypotenuse of a 45-45-90 triangle with legs measuring 18 cm.

Length of hypotenuse = √2 * length of each leg = √2 * 18 cm = 18√2 cm

  • Find the length of each leg of a 45-45-90 triangle with a hypotenuse measuring 24 cm.

Length of each leg = length of hypotenuse / √2 = 24 cm / √2 = 24√2 / 2 cm

  • Find the length of the hypotenuse of a 45-45-90 triangle with legs measuring 22 cm.

Length of hypotenuse = √2 * length of each leg = √2 * 22 cm = 22√2 cm

Conclusion

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

A: A 45-45-90 triangle is a special right triangle with two 45-degree angles and one 90-degree angle. The two 45-degree angles are equal, and the hypotenuse is the side opposite the 90-degree angle.

Q: What are the properties of a 45-45-90 triangle?

A: The properties of a 45-45-90 triangle are as follows:

  • The two legs are equal in length.
  • The hypotenuse is √2 times the length of each leg.
  • The two 45-degree angles are equal.
  • The 90-degree angle is opposite the hypotenuse.

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

A: To find the length of the hypotenuse, you can multiply the length of each leg by √2.

Q: What is the formula for finding the length of the hypotenuse of a 45-45-90 triangle?

A: The formula for finding the length of the hypotenuse is:

Length of hypotenuse = √2 * length of each leg

Q: Can I use the properties of a 45-45-90 triangle to find the length of each leg?

A: Yes, you can use the properties of a 45-45-90 triangle to find the length of each leg. To do this, you can divide the length of the hypotenuse by √2.

Q: What is the formula for finding the length of each leg of a 45-45-90 triangle?

A: The formula for finding the length of each leg is:

Length of each leg = length of hypotenuse / √2

Q: Can I use a 45-45-90 triangle to solve other types of problems?

A: Yes, you can use a 45-45-90 triangle to solve other types of problems. For example, you can use a 45-45-90 triangle to find the length of the hypotenuse of a right triangle with a given angle.

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

A: Some common applications of 45-45-90 triangles include:

  • Building design and construction
  • Engineering and architecture
  • Physics and mathematics
  • Computer graphics and game development

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 enter the length of each leg and multiply it by √2.

Q: What are some common mistakes to avoid when working with 45-45-90 triangles?

A: Some common mistakes to avoid when working with 45-45-90 triangles include:

  • Forgetting to multiply the length of each leg by √2 to find the length of the hypotenuse.
  • Forgetting to divide the length of the hypotenuse by √2 to find the length of each leg.
  • Not using the correct formula for finding the length of the hypotenuse or each leg.

Conclusion

In this article, we answered some frequently asked questions about 45-45-90 triangles. We covered topics such as the properties of a 45-45-90 triangle, how to find the length of the hypotenuse, and common applications of 45-45-90 triangles. We also discussed some common mistakes to avoid when working with 45-45-90 triangles.