Calculating the Length of One Leg

Oct 13, 2025 by wbrum.org 37 views

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 relationship between the legs and the hypotenuse of a 45°-45°-90° triangle, and explore how to calculate the length of one of the legs when the hypotenuse is known.

Properties of a 45°-45°-90° Triangle

A 45°-45°-90° triangle is a right-angled triangle with two equal legs and a hypotenuse that is √2 times the length of each leg. This unique property makes it an essential concept in geometry and trigonometry. The ratio of the sides of a 45°-45°-90° triangle is 1:1:√2, where the two legs are equal in length and the hypotenuse is √2 times the length of each leg.

Calculating the Length of One Leg

Given that the hypotenuse of a 45°-45°-90° triangle measures 24 inches, we can use the properties of the triangle to calculate the length of one of the legs. Since the ratio of the sides is 1:1:√2, we can set up a proportion to find the length of one leg.

Let x be the length of one leg. Then, the hypotenuse is √2 times the length of each leg, which is x√2. Since the hypotenuse is given as 24 inches, we can set up the equation:

x√2 = 24

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

x = 24 / √2

x = 24√2 / (√2 × √2)

x = 24√2 / 2

x = 12√2

Therefore, the length of one of the legs of the triangle is 12√2 inches.

Conclusion

In conclusion, the length of one of the legs of a 45°-45°-90° triangle can be calculated using the properties of the triangle. Given that the hypotenuse measures 24 inches, we can use the ratio of the sides to find the length of one leg, which is 12√2 inches. This unique property of the 45°-45°-90° triangle makes it an essential concept in geometry and trigonometry.

Frequently Asked Questions

What is the ratio of the sides of a 45°-45°-90° triangle? The ratio of the sides of a 45°-45°-90° triangle is 1:1:√2.
How can we calculate the length of one leg of a 45°-45°-90° triangle? We can use the properties of the triangle to set up a proportion and solve for the length of one leg.
What is the length of one leg of a 45°-45°-90° triangle with a hypotenuse of 24 inches? The length of one leg of a 45°-45°-90° triangle with a hypotenuse of 24 inches is 12√2 inches.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Trigonometry: A Unit Circle Approach" by Michael Sullivan
[3] "Mathematics for the Nonmathematician" by Morris Kline

Introduction

In our previous article, we explored the properties of a 45°-45°-90° triangle and how to calculate the length of one of the legs when the hypotenuse is known. In this article, we will address some of the most frequently asked questions related to the hypotenuse of a 45°-45°-90° triangle.

Q&A

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

A: The ratio of the sides of a 45°-45°-90° triangle is 1:1:√2, where the two legs are equal in length and the hypotenuse is √2 times the length of each leg.

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

A: We can use the properties of the triangle to set up a proportion and solve for the length of one leg. Given that the hypotenuse is √2 times the length of each leg, we can set up the equation:

x√2 = hypotenuse

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

x = hypotenuse / √2

Q: What is the length of one leg of a 45°-45°-90° triangle with a hypotenuse of 24 inches?

A: The length of one leg of a 45°-45°-90° triangle with a hypotenuse of 24 inches is 12√2 inches.

Q: Can we use the Pythagorean theorem to calculate the length of one leg of a 45°-45°-90° triangle?

A: Yes, we can use the Pythagorean theorem to calculate the length of one leg of a 45°-45°-90° triangle. The Pythagorean theorem states that:

a² + b² = c²

where a and b are the lengths of the legs and c is the length of the hypotenuse. Since the two legs are equal in length, we can set up the equation:

x² + x² = c²

Combine like terms:

2x² = c²

Divide both sides of the equation by 2:

x² = c² / 2

Take the square root of both sides of the equation:

x = √(c² / 2)

x = √(c / √2)

x = c / √2

This is the same equation we derived earlier using the properties of the triangle.

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 is that the hypotenuse is √2 times the length of each leg. This means that if we know the length of one leg, we can calculate the length of the hypotenuse by multiplying the length of the leg by √2.

Q: Can we use the properties of a 45°-45°-90° triangle to calculate the length of the hypotenuse?

A: Yes, we can use the properties of a 45°-45°-90° triangle to calculate the length of the hypotenuse. Given that the length of one leg is x, we can set up the equation:

x√2 = hypotenuse

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

hypotenuse = x√2

Conclusion

In conclusion, the hypotenuse of a 45°-45°-90° triangle is √2 times the length of each leg. We can use the properties of the triangle to calculate the length of one leg or the hypotenuse. By understanding the relationship between the legs and the hypotenuse, we can solve problems involving 45°-45°-90° triangles.

Frequently Asked Questions

What is the ratio of the sides of a 45°-45°-90° triangle?
How can we calculate the length of one leg of a 45°-45°-90° triangle?
What is the length of one leg of a 45°-45°-90° triangle with a hypotenuse of 24 inches?
Can we use the Pythagorean theorem to calculate the length of one leg of a 45°-45°-90° triangle?
What is the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle?

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Trigonometry: A Unit Circle Approach" by Michael Sullivan
[3] "Mathematics for the Nonmathematician" by Morris Kline