The Hypotenuse Of A $45 {\circ}-45 {\circ}-90^{\circ}$ Triangle Measures $22 \sqrt{2}$ Units.What Is The Length Of One Leg Of The Triangle?A. 11 Units B. $ 11 2 11 \sqrt{2} 11 2 [/tex] Units C. 22 Units D. $22

Mar 12, 2025 by ADMIN 220 views

The Hypotenuse of a 45°-45°-90° Triangle: Understanding the Relationship Between Legs and Hypotenuse

Introduction

In the realm of geometry, triangles are fundamental shapes that have been studied extensively. One of the most common types of triangles is the 45°-45°-90° triangle, also known as an 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 delve into the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle and explore how to find the length of one leg when the hypotenuse is given.

Understanding the 45°-45°-90° Triangle

A 45°-45°-90° triangle is a special type of right triangle that has two equal acute angles, each measuring 45°. The third angle, which is the right angle, measures 90°. This type of triangle has several unique properties that make it an essential concept in geometry.

One of the key properties of a 45°-45°-90° triangle is that the two legs are equal in length. This means that if one leg is x units long, the other leg is also x units long. The hypotenuse, which is the side opposite the right angle, is equal to the leg length multiplied by the square root of 2.

The Relationship Between Legs and Hypotenuse

The relationship between the legs and the hypotenuse of a 45°-45°-90° triangle can be expressed mathematically as:

c = a√2

where c is the length of the hypotenuse, a is the length of one leg, and √2 is the square root of 2.

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

Finding the Length of One Leg

Now that we have established the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle, we can use this information to find the length of one leg when the hypotenuse is given.

In this problem, we are given that the hypotenuse measures 22√2 units. We can use the equation c = a√2 to find the length of one leg.

Step 1: Write down the equation

c = a√2

Step 2: Plug in the value of c

22√2 = a√2

Step 3: Divide both sides by √2

a = 22

Step 4: Simplify the equation

a = 22

The final answer is that the length of one leg of the triangle is 22 units.

However, this is not the only possible answer. We can also express the length of one leg in terms of the square root of 2.

Alternative Solution

We can also express the length of one leg in terms of the square root of 2.

c = a√2

22√2 = a√2

a = 22

a = 22/√2

a = 22√2/√2

a = 22

a = 22/√2

a = 22√2/√2

a = 22√

Q&A: Frequently Asked Questions About 45°-45°-90° Triangles

In this article, we will answer some of the most frequently asked questions about 45°-45°-90° triangles, including the relationship between the legs and the hypotenuse.

Q: What is a 45°-45°-90° triangle?

A: A 45°-45°-90° triangle is a special type of right triangle that has two equal acute angles, each measuring 45°. The third angle, which is the right angle, measures 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 mathematically as:

c = a√2

where c is the length of the hypotenuse, a is the length of one leg, and √2 is the square root of 2.

Q: How do I find the length of one leg when the hypotenuse is given?

A: To find the length of one leg when the hypotenuse is given, you can use the equation c = a√2. Simply divide the length of the hypotenuse by the square root of 2 to find the length of one leg.

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

A: To find the length of one leg, we can use the equation c = a√2. Plugging in the value of c, we get:

22√2 = a√2

Dividing both sides by √2, we get:

a = 22

So, the length of one leg of the triangle is 22 units.

Q: Can I express the length of one leg in terms of the square root of 2?

A: Yes, you can express the length of one leg in terms of the square root of 2. To do this, we can rewrite the equation c = a√2 as:

a = c/√2

Plugging in the value of c, we get:

a = 22√2/√2

Simplifying the equation, we get:

a = 22

So, 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: 45°-45°-90° triangles are often used in building design to create symmetrical and aesthetically pleasing structures.
Engineering: 45°-45°-90° triangles are used in engineering to design and build bridges, buildings, and other structures.
Art: 45°-45°-90° triangles are used in art to create symmetrical and balanced compositions.

Q: How can I use 45°-45°-90° triangles in my daily life?

A: 45°-45°-90° triangles can be used in many ways in your daily life, including:

Measuring rooms: 45°-45°-90° triangles can be used to measure the length and width of rooms.
Designing furniture: 45°-45°-90° triangles can be used to design and build furniture, such as tables and chairs.
Creating art: 45°-45°-90° triangles can be used to create symmetrical and balanced compositions in art.

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 the hypotenuse with the length of one leg.
Not using the correct equation to find the length of one leg.
Not simplifying the equation to find the length of one leg.

By avoiding these common mistakes, you can ensure that you are working with 45°-45°-90° triangles correctly and accurately.