If The Legs Of An Isosceles Right Triangle Have A Length Of $15 \sqrt{2}$ Feet, What Is The Length Of The Hypotenuse?A. 7.5 Feet B. $15 \sqrt{2}$ Feet C. \$15 \sqrt{3}$[/tex\] Feet D. 30 Feet

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If the Legs of an Isosceles Right Triangle Have a Length of $15 \sqrt{2}$ Feet, What is the Length of the Hypotenuse?

Understanding the Basics of an Isosceles Right Triangle

An isosceles right triangle is a special type of right triangle where the two legs are equal in length, and the angle between them is 90 degrees. This type of triangle is also known as a 45-45-90 triangle, as the two acute angles are equal and measure 45 degrees each. In this type of triangle, the hypotenuse is always √2 times the length of a leg.

Calculating the Length of the Hypotenuse

Given that the legs of the isosceles right triangle have a length of $15 \sqrt{2}$ feet, we can use the properties of a 45-45-90 triangle to calculate the length of the hypotenuse. In a 45-45-90 triangle, the hypotenuse is equal to √2 times the length of a leg. Therefore, we can calculate the length of the hypotenuse as follows:

Hypotenuse = √2 × leg length Hypotenuse = √2 × $15 \sqrt{2}$ Hypotenuse = $15 \sqrt{2}$ × √2 Hypotenuse = $15 \times 2$ Hypotenuse = 30 feet

Analyzing the Options

Now that we have calculated the length of the hypotenuse, let's analyze the options provided:

A. 7.5 feet - This is incorrect, as the length of the hypotenuse is not 7.5 feet.

B. $15 \sqrt{2}$ feet - This is incorrect, as the length of the hypotenuse is not equal to the length of a leg.

C. $15 \sqrt{3}$ feet - This is incorrect, as the length of the hypotenuse is not equal to $15 \sqrt{3}$ feet.

D. 30 feet - This is correct, as the length of the hypotenuse is indeed 30 feet.

Conclusion

In conclusion, if the legs of an isosceles right triangle have a length of $15 \sqrt{2}$ feet, the length of the hypotenuse is 30 feet. This is because the hypotenuse of a 45-45-90 triangle is equal to √2 times the length of a leg, and in this case, the length of a leg is $15 \sqrt{2}$ feet.

Properties of a 45-45-90 Triangle

A 45-45-90 triangle is a special type of right triangle where the two legs are equal in length, and the angle between them is 90 degrees. The properties of a 45-45-90 triangle are as follows:

  • The two legs are equal in length.
  • The angle between the two legs is 90 degrees.
  • The hypotenuse is equal to √2 times the length of a leg.
  • The ratio of the lengths of the sides is 1:1:√2.

Real-World Applications of 45-45-90 Triangles

45-45-90 triangles have many real-world applications, including:

  • Building design: 45-45-90 triangles are often used in building design to create strong and stable structures.
  • Engineering: 45-45-90 triangles are used in engineering to calculate the stresses and strains on structures.
  • Physics: 45-45-90 triangles are used in physics to calculate the motion of objects.

Conclusion

In conclusion, 45-45-90 triangles are an important concept in mathematics and have many real-world applications. Understanding the properties of a 45-45-90 triangle is essential for solving problems involving right triangles.
Q&A: If the Legs of an Isosceles Right Triangle Have a Length of $15 \sqrt{2}$ Feet, What is the Length of the Hypotenuse?

Frequently Asked Questions

Q: What is an isosceles right triangle?

A: An isosceles right triangle is a special type of right triangle where the two legs are equal in length, and the angle between them is 90 degrees. This type of triangle is also known as a 45-45-90 triangle, as the two acute angles are equal and measure 45 degrees each.

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 hypotenuse is equal to √2 times the length of a leg. This means that if you know the length of a leg, you can calculate the length of the hypotenuse by multiplying it by √2.

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

A: To calculate the length of the hypotenuse of a 45-45-90 triangle, you can use the formula:

Hypotenuse = √2 × leg length

For example, if the length of a leg is $15 \sqrt{2}$ feet, the length of the hypotenuse would be:

Hypotenuse = √2 × $15 \sqrt{2}$ Hypotenuse = $15 \sqrt{2}$ × √2 Hypotenuse = $15 \times 2$ Hypotenuse = 30 feet

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

A: The ratio of the lengths of the sides of a 45-45-90 triangle is 1:1:√2. This means that the length of a leg is equal to the length of the other leg, and the length of the hypotenuse is √2 times the length of a leg.

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

A: 45-45-90 triangles have many real-world applications, including:

  • Building design: 45-45-90 triangles are often used in building design to create strong and stable structures.
  • Engineering: 45-45-90 triangles are used in engineering to calculate the stresses and strains on structures.
  • Physics: 45-45-90 triangles are used in physics to calculate the motion of objects.

Q: Can I use a 45-45-90 triangle to solve problems involving right triangles?

A: Yes, you can use a 45-45-90 triangle to solve problems involving right triangles. By understanding the properties of a 45-45-90 triangle, you can apply them to solve problems involving right triangles.

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:

  • Confusing the length of a leg with the length of the hypotenuse.
  • Failing to recognize the properties of a 45-45-90 triangle.
  • Not using the correct formula to calculate the length of the hypotenuse.

Conclusion

In conclusion, 45-45-90 triangles are an important concept in mathematics and have many real-world applications. By understanding the properties of a 45-45-90 triangle, you can apply them to solve problems involving right triangles.