Relationship Between Legs and Hypotenuse

Oct 13, 2025 by wbrum.org 141 views

Introduction

In the realm of geometry, triangles are a fundamental concept that has been studied extensively. One of the most common types of triangles is the 45°-45°-90° triangle, also known as the isosceles right triangle. This type of triangle has two equal legs and a hypotenuse that is equal to the leg length multiplied by the square root of 2. In this article, we will explore the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle and use this knowledge to solve a problem involving the length of one leg.

Properties of a 45°-45°-90° Triangle

A 45°-45°-90° triangle is a special type of right triangle that has two equal legs and a hypotenuse that is equal to the leg length multiplied by the square root of 2. The properties of this triangle can be summarized as follows:

The two legs are equal in length.
The hypotenuse is equal to the leg length multiplied by the square root of 2.
The angles of the triangle are 45°, 45°, and 90°.

Relationship Between Legs and Hypotenuse

The relationship between the legs and the hypotenuse of a 45°-45°-90° triangle can be expressed using the following formula:

Hypotenuse = leg length × √2

This formula shows that the hypotenuse is equal to the leg length multiplied by the square root of 2. This means that if we know the length of one leg, we can calculate the length of the hypotenuse by multiplying the leg length by the square root of 2.

Problem: Finding the Length of One Leg

The problem states that the hypotenuse of a 45°-45°-90° triangle measures 22√2 units. We are asked to find the length of one leg of the triangle. To solve this problem, we can use the formula for the relationship between the legs and the hypotenuse:

Hypotenuse = leg length × √2

We are given that the hypotenuse measures 22√2 units. We can set up an equation using this information:

22√2 = leg length × √2

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

leg length = 22√2 / √2

leg length = 22

This means that the length of one leg of the triangle is 22 units.

Conclusion

In this article, we explored the properties of a 45°-45°-90° triangle and the relationship between the legs and the hypotenuse. We used this knowledge to solve a problem involving the length of one leg of a 45°-45°-90° triangle. The problem stated that the hypotenuse measures 22√2 units, and we were able to find the length of one leg by using the formula for the relationship between the legs and the hypotenuse. The length of one leg of the triangle is 22 units.

Frequently Asked Questions

What is a 45°-45°-90° triangle?
What are the properties of a 45°-45°-90° triangle?
What is the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle?
How can we use the formula for the relationship between the legs and the hypotenuse to solve problems involving 45°-45°-90° triangles?

References

[1] "Geometry: A Comprehensive Introduction." By Dan Pedoe. Dover Publications, 2012.
[2] "Mathematics for the Nonmathematician." By Morris Kline. Dover Publications, 2013.
[3] "Geometry: A Guide for Teachers." By David A. Smith. Pearson Education, 2015.

Additional Resources

[1] Khan Academy. "Geometry." https://www.khanacademy.org/math/geometry
[2] Math Open Reference. "Geometry." https://www.mathopenref.com/geometry.html
[3] Wolfram MathWorld. "Geometry." https://mathworld.wolfram.com/Geometry.html

Q&A: Understanding 45°-45°-90° Triangles

Q: What is a 45°-45°-90° triangle?

A: A 45°-45°-90° triangle, also known as an isosceles right triangle, is a special type of right triangle that has two equal legs and a hypotenuse that is equal to the leg length multiplied by the square root of 2.

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

A: The properties of a 45°-45°-90° triangle can be summarized as follows:

The two legs are equal in length.
The hypotenuse is equal to the leg length multiplied by the square root of 2.
The angles of the triangle are 45°, 45°, and 90°.

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

A: The relationship between the legs and the hypotenuse of a 45°-45°-90° triangle can be expressed using the following formula:

Hypotenuse = leg length × √2

This formula shows that the hypotenuse is equal to the leg length multiplied by the square root of 2.

Q: How can we use the formula for the relationship between the legs and the hypotenuse to solve problems involving 45°-45°-90° triangles?

A: To use the formula for the relationship between the legs and the hypotenuse to solve problems involving 45°-45°-90° triangles, we can follow these steps:

Identify the length of the hypotenuse.
Use the formula to set up an equation.
Solve for the leg length.

Q: What is the length of one leg of a 45°-45°-90° triangle if the hypotenuse measures 22√2 units?

A: To find the length of one leg of a 45°-45°-90° triangle if the hypotenuse measures 22√2 units, we can use the formula for the relationship between the legs and the hypotenuse:

Hypotenuse = leg length × √2

We can set up an equation using this information:

22√2 = leg length × √2

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

leg length = 22√2 / √2

leg length = 22

This means that the length of one leg of the triangle is 22 units.

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 and construction
Engineering and architecture
Art and design
Physics and mathematics

Q: How can we use 45°-45°-90° triangles to solve problems in real-world applications?

A: To use 45°-45°-90° triangles to solve problems in real-world applications, we can follow these steps:

Identify the problem and the relevant information.
Draw a diagram of the problem using a 45°-45°-90° triangle.
Use the formula for the relationship between the legs and the hypotenuse to set up an equation.
Solve for the unknown variable.

Conclusion

In this article, we answered some frequently asked questions about 45°-45°-90° triangles. We discussed the properties of these triangles, the relationship between the legs and the hypotenuse, and how to use the formula for the relationship between the legs and the hypotenuse to solve problems involving 45°-45°-90° triangles. We also explored some real-world applications of 45°-45°-90° triangles and how to use them to solve problems in real-world applications.

Frequently Asked Questions

What is a 45°-45°-90° triangle?
What are the properties of a 45°-45°-90° triangle?
What is the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle?
How can we use the formula for the relationship between the legs and the hypotenuse to solve problems involving 45°-45°-90° triangles?

References

[1] "Geometry: A Comprehensive Introduction." By Dan Pedoe. Dover Publications, 2012.
[2] "Mathematics for the Nonmathematician." By Morris Kline. Dover Publications, 2013.
[3] "Geometry: A Guide for Teachers." By David A. Smith. Pearson Education, 2015.

Additional Resources

[1] Khan Academy. "Geometry." https://www.khanacademy.org/math/geometry
[2] Math Open Reference. "Geometry." https://www.mathopenref.com/geometry.html
[3] Wolfram MathWorld. "Geometry." https://mathworld.wolfram.com/Geometry.html