Confusing the legs and the hypotenuse

Oct 13, 2025 by pinkandtrash.com 19 views

Solving a Classic Geometry Problem: The $$45^{\circ}-45{\circ}-90^{\circ}$$ Triangle

In geometry, a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is a special right-angled triangle with two equal angles, each measuring $$45^{\circ}$$. This type of triangle has several unique properties that make it an essential concept in mathematics. In this article, we will explore the properties of the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle and use it to solve a classic problem involving a triangle with equal leg lengths.

A $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle has several key properties that make it distinct from other types of triangles. These properties include:

Two equal angles: The two acute angles in a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle are equal, each measuring $$45^{\circ}$$.
One right angle: The third angle in a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is a right angle, measuring $$90^{\circ}$$.
Equal leg lengths: The two legs of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle are equal in length.
**Hypotenuse is $$\sqrt2}$$ times the leg length** The length of the hypotenuse of a $$45^{\circ-45^{\circ}-90{\circ}$$ triangle is $$\sqrt{2}$$ times the length of each leg.

Now that we have explored the properties of the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle, let's use this knowledge to solve the problem presented in the introduction.

Problem Statement

Each leg of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle measures 14 cm. What is the length of the hypotenuse?

Solution

To solve this problem, we can use the property of the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle that states the hypotenuse is $$\sqrt{2}$$ times the leg length. Since each leg measures 14 cm, we can multiply this length by $$\sqrt{2}$$ to find the length of the hypotenuse.

import math

leg_length = 14  # cm
hypotenuse_length = leg_length * math.sqrt(2)
print("The length of the hypotenuse is {:.2f} cm".format(hypotenuse_length))

Answer

The length of the hypotenuse is $$14 \sqrt{2}$$ cm.

In this article, we explored the properties of the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle and used this knowledge to solve a classic problem involving a triangle with equal leg lengths. We demonstrated how to use the property of the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle that states the hypotenuse is $$\sqrt{2}$$ times the leg length to find the length of the hypotenuse. This problem is a great example of how geometry can be used to solve real-world problems and is an essential concept in mathematics.

[1] "Geometry: A Comprehensive Introduction". By Dan Pedoe. Published by Dover Publications.
[2] "Mathematics for Computer Science". By Eric Lehman, F Thomson Leighton, and Albert R Meyer. Published by MIT OpenCourseWare.

[1] Khan Academy: Geometry
[2] MIT OpenCourseWare: Mathematics for Computer Science
[3] Wolfram MathWorld: $$45^{\circ}-45\circ}-90^{\circ}$$ Triangle
**Frequently Asked Questions $$45^{\circ-45^{\circ}-90{\circ}$$ Triangle**

In our previous article, we explored the properties of the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle and used it to solve a classic problem involving a triangle with equal leg lengths. In this article, we will answer some of the most frequently asked questions about the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle.

Q: What is the relationship between the legs and the hypotenuse of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle?

A: The hypotenuse of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is $$\sqrt{2}$$ times the length of each leg.

Q: How do I find the length of the hypotenuse of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle?

A: To find the length of the hypotenuse, you can multiply the length of each leg by $$\sqrt{2}$$.

Q: What is the formula for the length of the hypotenuse of a $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle?

A: The formula for the length of the hypotenuse is:

h = l \sqrt{2}

where $$h$$ is the length of the hypotenuse and $$l$$ is the length of each leg.

Q: Can I use the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle to solve problems involving right triangles?

A: Yes, the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is a special type of right triangle that can be used to solve problems involving right triangles.

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

A: The $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle has many real-world applications, including:

Construction: The $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is used in construction to calculate the length of rafters and other building components.
Engineering: The $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is used in engineering to calculate the length of beams and other structural components.
Design: The $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is used in design to create symmetrical and balanced compositions.

Q: Can I use the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle to solve problems involving oblique triangles?

A: No, the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle is only used to solve problems involving right triangles.

Q: What are some common mistakes to avoid when working with the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle?

A: Some common mistakes to avoid when working with the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle include:

Confusing the legs and the hypotenuse: Make sure to identify the legs and the hypotenuse correctly before solving a problem.
Using the wrong formula: Use the correct formula for the length of the hypotenuse, which is $$h = l \sqrt{2}$$.
Not checking units: Make sure to check the units of the answer to ensure that they are correct.

In this article, we answered some of the most frequently asked questions about the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle. We covered topics such as the relationship between the legs and the hypotenuse, how to find the length of the hypotenuse, and common mistakes to avoid when working with the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle. We hope that this article has been helpful in clarifying any questions you may have had about the $$45^{\circ}-45{\circ}-90^{\circ}$$ triangle.

[1] "Geometry: A Comprehensive Introduction". By Dan Pedoe. Published by Dover Publications.
[2] "Mathematics for Computer Science". By Eric Lehman, F Thomson Leighton, and Albert R Meyer. Published by MIT OpenCourseWare.
[3] Khan Academy: Geometry
[4] MIT OpenCourseWare: Mathematics for Computer Science
[5] Wolfram MathWorld: $$45^{\circ}-45{\circ}-90^{\circ}$$ Triangle