The Hypotenuse Of A $45 {\circ}-45 {\circ}-90^{\circ}$ Triangle Measures 18 Cm. Determine The Length Of Each Leg Of The Triangle. A. $ 9 2 9 \sqrt{2} 9 2 ​ $ Cm$ B. 18 Cm C. $18 \sqrt{2}$ Cm$

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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 world of 45°-45°-90° triangles and explore the relationship between the hypotenuse and the legs of such a triangle.

Understanding the 45°-45°-90° Triangle

A 45°-45°-90° triangle is a right-angled triangle with two acute angles, each measuring 45°. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. In a 45°-45°-90° triangle, the legs are congruent, meaning they have the same length.

The Relationship Between the Hypotenuse and the Legs

In a 45°-45°-90° triangle, the hypotenuse is √2 times the length of each leg. This relationship can be expressed mathematically as:

hypotenuse = leg × √2

Determining the Length of Each Leg

Given that the hypotenuse of a 45°-45°-90° triangle measures 18 cm, we can use the relationship between the hypotenuse and the legs to determine the length of each leg.

Let's denote the length of each leg as x. Using the relationship:

hypotenuse = leg × √2

We can substitute the given value of the hypotenuse (18 cm) and solve for x:

18 = x × √2

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

x = 18 ÷ √2

x = 18 × √2/√2

x = 18 × √2/√2 × √2/√2

x = 18 × √2/2

x = 9 × √2

Therefore, the length of each leg of the triangle is 9√2 cm.

Conclusion

In conclusion, we have explored the relationship between the hypotenuse and the legs of a 45°-45°-90° triangle. Using the given value of the hypotenuse (18 cm), we have determined that the length of each leg of the triangle is 9√2 cm. This result is consistent with the mathematical relationship between the hypotenuse and the legs of a 45°-45°-90° triangle.

Frequently Asked Questions

  • Q: What is the relationship between the hypotenuse and the legs of a 45°-45°-90° triangle? A: The hypotenuse is √2 times the length of each leg.
  • Q: How can we determine the length of each leg of a 45°-45°-90° triangle? A: We can use the relationship between the hypotenuse and the legs to solve for the length of each leg.
  • Q: What is the length of each leg of a 45°-45°-90° triangle with a hypotenuse of 18 cm? A: The length of each leg is 9√2 cm.

Additional Resources

  • For more information on 45°-45°-90° triangles, please refer to the following resources:
  • Khan Academy: 45°-45°-90° Triangle
  • Math Open Reference: 45°-45°-90° Triangle
  • Wolfram MathWorld: 45°-45°-90° Triangle

Final Thoughts

In conclusion, the 45°-45°-90° triangle is a unique and fascinating concept in geometry. By understanding the relationship between the hypotenuse and the legs of such a triangle, we can solve problems and determine the length of each leg. We hope that this article has provided valuable insights and information on the 45°-45°-90° triangle.

Introduction

In our previous article, we explored the relationship between the hypotenuse and the legs of a 45°-45°-90° triangle. In this article, we will answer some frequently asked questions about 45°-45°-90° triangles.

Q&A

Q: What is a 45°-45°-90° triangle?

A: A 45°-45°-90° triangle is a right-angled triangle with two acute angles, each measuring 45°. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

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

A: The hypotenuse is √2 times the length of each leg.

Q: How can we determine the length of each leg of a 45°-45°-90° triangle?

A: We can use the relationship between the hypotenuse and the legs to solve for the length of each leg.

Q: What is the length of each leg of a 45°-45°-90° triangle with a hypotenuse of 18 cm?

A: The length of each leg is 9√2 cm.

Q: Are the legs of a 45°-45°-90° triangle always equal?

A: Yes, the legs of a 45°-45°-90° triangle are always equal.

Q: Can a 45°-45°-90° triangle have a hypotenuse of 0 cm?

A: No, a 45°-45°-90° triangle cannot have a hypotenuse of 0 cm, as the hypotenuse is always greater than 0 cm.

Q: What is the sum of the lengths of the legs of a 45°-45°-90° triangle?

A: The sum of the lengths of the legs of a 45°-45°-90° triangle is equal to the length of the hypotenuse.

Q: Can a 45°-45°-90° triangle be isosceles?

A: Yes, a 45°-45°-90° triangle is isosceles, as the two legs are equal in length.

Q: Can a 45°-45°-90° triangle be equilateral?

A: No, a 45°-45°-90° triangle cannot be equilateral, as the angles are not all equal.

Conclusion

In conclusion, we have answered some frequently asked questions about 45°-45°-90° triangles. We hope that this article has provided valuable insights and information on this unique and fascinating concept in geometry.

Additional Resources

  • For more information on 45°-45°-90° triangles, please refer to the following resources:
  • Khan Academy: 45°-45°-90° Triangle
  • Math Open Reference: 45°-45°-90° Triangle
  • Wolfram MathWorld: 45°-45°-90° Triangle

Final Thoughts

In conclusion, the 45°-45°-90° triangle is a unique and fascinating concept in geometry. By understanding the relationship between the hypotenuse and the legs of such a triangle, we can solve problems and determine the length of each leg. We hope that this article has provided valuable insights and information on the 45°-45°-90° triangle.

Common Misconceptions

  • Many people believe that a 45°-45°-90° triangle is always a right triangle, but this is not true. A 45°-45°-90° triangle is a right triangle, but not all right triangles are 45°-45°-90° triangles.
  • Some people believe that the hypotenuse of a 45°-45°-90° triangle is always equal to the sum of the lengths of the legs, but this is not true. The hypotenuse is √2 times the length of each leg.

Real-World Applications

  • 45°-45°-90° triangles are used in architecture to design buildings and bridges.
  • 45°-45°-90° triangles are used in engineering to design machines and mechanisms.
  • 45°-45°-90° triangles are used in art to create geometric patterns and designs.

Final Tips

  • When working with 45°-45°-90° triangles, always remember that the hypotenuse is √2 times the length of each leg.
  • When solving problems involving 45°-45°-90° triangles, always use the relationship between the hypotenuse and the legs to determine the length of each leg.
  • When creating geometric patterns and designs, always use 45°-45°-90° triangles to create unique and fascinating designs.