In A 45 ∘ − 45 ∘ − 90 ∘ 45^{\circ}-45^{\circ}-90^{\circ} 4 5 ∘ − 4 5 ∘ − 9 0 ∘ Triangle, What Is The Length Of The Hypotenuse When The Length Of One Of The Legs Is 8 In.?A. 2 8 2 \sqrt{8} 2 8 ​ In. B. 8 8 8 \sqrt{8} 8 8 ​ In. C. 8 \sqrt{8} 8 ​ In. D. 8 2 8 \sqrt{2} 8 2 ​

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A 45∘−45∘−90∘45{\circ}-45\circ}-90^{\circ}45∘−90∘ triangle is a special right-angled triangle with two equal angles, each measuring 45∘45^{\circ. This unique configuration results in two equal sides, known as legs, and a third side, known as the hypotenuse, which is opposite the right angle. In this article, we will explore the properties of the 45∘−45∘−90∘45{\circ}-45{\circ}-90^{\circ} triangle and determine the length of the hypotenuse when one of the legs is 8 inches.

Properties of the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} Triangle

The 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle has several distinct properties that make it an essential concept in geometry and trigonometry. Some of the key properties include:

  • Two equal legs: The two sides that meet at a 45∘45^{\circ} angle are equal in length.
  • Hypotenuse is 2\sqrt{2} times the length of a leg: The length of the hypotenuse is 2\sqrt{2} times the length of one of the legs.
  • Right angle: The triangle has a right angle, which is 90∘90^{\circ}.

Calculating the Length of the Hypotenuse

To calculate the length of the hypotenuse, we can use the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. However, since the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle has two equal legs, we can simplify the calculation by using the property that the hypotenuse is 2\sqrt{2} times the length of a leg.

Given that one of the legs is 8 inches, we can calculate the length of the hypotenuse as follows:

  • Length of the hypotenuse: 8×2=828 \times \sqrt{2} = 8\sqrt{2} inches

Conclusion

In conclusion, the length of the hypotenuse in a 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle is 828\sqrt{2} inches when one of the legs is 8 inches. This result is obtained by using the property that the hypotenuse is 2\sqrt{2} times the length of a leg.

Answer

The correct answer is:

  • D. 828 \sqrt{2}

Additional Examples

To further illustrate the concept, let's consider a few more examples:

  • Example 1: If one of the legs is 10 inches, what is the length of the hypotenuse?
    • Length of the hypotenuse: 10×2=10210 \times \sqrt{2} = 10\sqrt{2} inches
  • Example 2: If one of the legs is 15 inches, what is the length of the hypotenuse?
    • Length of the hypotenuse: 15×2=15215 \times \sqrt{2} = 15\sqrt{2} inches

Real-World Applications

The 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle has numerous real-world applications in various fields, including:

  • Architecture: The 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle is used in the design of buildings, bridges, and other structures.
  • Engineering: The triangle is used in the design of mechanical systems, electrical systems, and other engineering applications.
  • Art and Design: The 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle is used in the creation of geometric patterns and designs.

Conclusion

Q: What is a 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle?

A: A 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle is a special right-angled triangle with two equal angles, each measuring 45∘45^{\circ}. This unique configuration results in two equal sides, known as legs, and a third side, known as the hypotenuse, which is opposite the right angle.

Q: What are the properties of a 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle?

A: The 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle has several distinct properties, including:

  • Two equal legs: The two sides that meet at a 45∘45^{\circ} angle are equal in length.
  • Hypotenuse is 2\sqrt{2} times the length of a leg: The length of the hypotenuse is 2\sqrt{2} times the length of one of the legs.
  • Right angle: The triangle has a right angle, which is 90∘90^{\circ}.

Q: How do I calculate the length of the hypotenuse in a 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle?

A: To calculate the length of the hypotenuse, you can use the property that the hypotenuse is 2\sqrt{2} times the length of a leg. For example, if one of the legs is 8 inches, the length of the hypotenuse would be 8×2=828 \times \sqrt{2} = 8\sqrt{2} inches.

Q: What are some real-world applications of the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle?

A: The 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle has numerous real-world applications in various fields, including:

  • Architecture: The 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle is used in the design of buildings, bridges, and other structures.
  • Engineering: The triangle is used in the design of mechanical systems, electrical systems, and other engineering applications.
  • Art and Design: The 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle is used in the creation of geometric patterns and designs.

Q: Can I use the Pythagorean theorem to calculate the length of the hypotenuse in a 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle?

A: Yes, you can use the Pythagorean theorem to calculate the length of the hypotenuse in a 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle. However, since the triangle has two equal legs, you can simplify the calculation by using the property that the hypotenuse is 2\sqrt{2} times the length of a leg.

Q: What are some common mistakes to avoid when working with 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangles?

A: Some common mistakes to avoid when working with 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangles include:

  • Not using the correct formula: Make sure to use the correct formula for calculating the length of the hypotenuse, which is 2\sqrt{2} times the length of a leg.
  • Not considering the properties of the triangle: Make sure to consider the properties of the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle, including the two equal legs and the right angle.

Q: How can I apply the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle in real-world problems?

A: You can apply the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle in real-world problems by using its properties and formulas to solve problems and make calculations. For example, you can use the triangle to design buildings, bridges, and other structures, or to create geometric patterns and designs.

Q: What are some advanced topics related to the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle?

A: Some advanced topics related to the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle include:

  • Trigonometry: The 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle is used in trigonometry to solve problems and make calculations.
  • Geometry: The triangle is used in geometry to create geometric patterns and designs.
  • Engineering: The triangle is used in engineering to design mechanical systems, electrical systems, and other engineering applications.

Conclusion

In conclusion, the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle is a fundamental concept in geometry and trigonometry. Its properties and applications make it an essential tool in various fields. By understanding the properties of the 45∘−45∘−90∘45^{\circ}-45^{\circ}-90^{\circ} triangle, you can solve problems and make calculations with ease.