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. $6 \sqrt{2}$ Cm C. 12 Cm D. $ 12 2 12 \sqrt{2} 12 2 ​ [/tex] Cm

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Introduction

In geometry, a 45-45-90 triangle is a special right-angled triangle with two equal acute angles, each measuring 45 degrees. This type of triangle has several unique properties that make it easier to solve. In this article, we will explore how to find the length of the hypotenuse of a 45-45-90 triangle when each leg measures 12 cm.

Understanding the Properties of a 45-45-90 Triangle

A 45-45-90 triangle has several key properties that make it easy to solve. The two acute angles are equal, each measuring 45 degrees, and the third angle is a right angle, measuring 90 degrees. The two legs of the triangle are also equal in length. In this case, each leg measures 12 cm.

The Relationship Between the Legs and the Hypotenuse

In a 45-45-90 triangle, the length of the hypotenuse is always equal to the length of a leg multiplied by the square root of 2. This can be expressed mathematically as:

c = a√2

where c is the length of the hypotenuse, and a is the length of a leg.

Applying the Formula to Find the Length of the Hypotenuse

Now that we have the formula, we can apply it to find the length of the hypotenuse. We know that each leg measures 12 cm, so we can plug this value into the formula:

c = 12√2

Simplifying the Expression

To simplify the expression, we can multiply 12 by √2:

c = 12√2 c = 12 × √2 c = 12 × 1.414 (approximately) c = 17.068 (approximately)

Conclusion

In conclusion, the length of the hypotenuse of a 45-45-90 triangle with each leg measuring 12 cm is approximately 17.068 cm. This can be expressed mathematically as:

c = 12√2

Answer

The correct answer is:

D. 12√2 cm

Why This Matters

Understanding the properties of a 45-45-90 triangle and how to find the length of the hypotenuse is important in geometry and trigonometry. It can be applied to a wide range of problems, from architecture to engineering. By mastering this concept, you can solve a variety of problems with ease.

Real-World Applications

The concept of a 45-45-90 triangle has many real-world applications. For example, in construction, builders use this type of triangle to calculate the length of rafters and roof beams. In engineering, it is used to design bridges and other structures. In architecture, it is used to design buildings and other structures.

Common Mistakes to Avoid

When solving a 45-45-90 triangle, there are several common mistakes to avoid. One mistake is to assume that the length of the hypotenuse is equal to the length of a leg. Another mistake is to forget to multiply the length of a leg by √2. By avoiding these mistakes, you can ensure that your calculations are accurate.

Tips and Tricks

Here are some tips and tricks to help you solve a 45-45-90 triangle:

  • Make sure to identify the type of triangle you are working with.
  • Use the formula c = a√2 to find the length of the hypotenuse.
  • Simplify the expression by multiplying the length of a leg by √2.
  • Check your calculations to ensure that they are accurate.

Conclusion

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

A: A 45-45-90 triangle is a special right-angled triangle with two equal acute angles, each measuring 45 degrees. This type of triangle has several unique properties that make it easier to solve.

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

A: The two acute angles of a 45-45-90 triangle are equal, each measuring 45 degrees. The third angle is a right angle, measuring 90 degrees. The two legs of the triangle are also equal in length.

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, you can use the formula c = a√2, where c is the length of the hypotenuse, and a is the length of a leg.

Q: What is the relationship between the legs and the hypotenuse of a 45-45-90 triangle?

A: In a 45-45-90 triangle, the length of the hypotenuse is always equal to the length of a leg multiplied by the square root of 2.

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

A: Yes, you can use the Pythagorean theorem to find the length of the hypotenuse of a 45-45-90 triangle. However, it is often easier to use the formula c = a√2.

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

A: 45-45-90 triangles have many real-world applications, including construction, engineering, and architecture. They are used to calculate the length of rafters and roof beams, design bridges and other structures, and design buildings and other structures.

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

A: Some common mistakes to avoid when solving 45-45-90 triangles include assuming that the length of the hypotenuse is equal to the length of a leg, and forgetting to multiply the length of a leg by √2.

Q: How can I simplify the expression c = a√2?

A: To simplify the expression c = a√2, you can multiply the length of a leg by √2. For example, if a = 12, then c = 12√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 enter the length of a leg and the calculator will give you the length of the hypotenuse.

Q: What is the significance of the square root of 2 in a 45-45-90 triangle?

A: The square root of 2 is a fundamental constant in mathematics that is used to describe the relationship between the legs and the hypotenuse of a 45-45-90 triangle.

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

A: While a 45-45-90 triangle is a special type of triangle, it can be used to solve other types of triangles by using trigonometric ratios and identities.

Q: How can I apply the concept of a 45-45-90 triangle to real-world problems?

A: The concept of a 45-45-90 triangle can be applied to a wide range of real-world problems, including construction, engineering, and architecture. By understanding the properties of a 45-45-90 triangle, you can solve problems with ease and accuracy.

Conclusion

In conclusion, 45-45-90 triangles are a fundamental concept in geometry and trigonometry that have many real-world applications. By understanding the properties of a 45-45-90 triangle, you can solve problems with ease and accuracy. Remember to use the formula c = a√2, simplify the expression, and check your calculations to ensure that they are accurate.