If Each Leg Of A 45-45-90 Degree Triangle Measures 14 Cm, What Is The Length Of The Hypotenuse?

In the realm of geometry, the 45-45-90 triangle holds a special place. Its unique angle measurements and side ratios make it a fundamental concept in trigonometry and various real-world applications. This article delves into the intricacies of the 45-45-90 triangle, focusing on determining the hypotenuse length when the legs are known. We'll explore the underlying principles, the relevant formulas, and practical examples to solidify your understanding. This article aims to provide a comprehensive guide, ensuring you grasp the concept thoroughly and can confidently tackle related problems. From students grappling with geometry homework to enthusiasts seeking a deeper understanding of mathematical relationships, this article caters to a wide audience. We'll break down the complexities into manageable steps, making the learning process engaging and effective. So, embark on this geometrical journey with us as we unravel the secrets of the 45-45-90 triangle and its hypotenuse.

Understanding the 45-45-90 Triangle

At its core, the 45-45-90 triangle is a right triangle, meaning it possesses one 90-degree angle. The other two angles are both 45 degrees, making it an isosceles triangle as well. This dual nature—right and isosceles—gives it special properties regarding side lengths. Specifically, the two legs (the sides opposite the 45-degree angles) are congruent, and the hypotenuse (the side opposite the 90-degree angle) has a length that's directly related to the leg length. Understanding these relationships is crucial for solving problems involving 45-45-90 triangles. This triangle isn't just a theoretical concept; it appears in numerous real-world scenarios, from construction and engineering to art and design. Its predictable side ratios make it a valuable tool in these fields. Imagine designing a roof with a specific slope or calculating the diagonal length of a square – the 45-45-90 triangle becomes your reliable companion. Furthermore, its presence in trigonometry is undeniable. It serves as a fundamental building block for understanding trigonometric functions and their applications. Mastery of the 45-45-90 triangle opens doors to more advanced concepts in mathematics and related disciplines.

The 45-45-90 Triangle Theorem: The Key to the Hypotenuse

The 45-45-90 triangle theorem is the cornerstone for calculating the hypotenuse. It elegantly states the relationship between the legs and the hypotenuse: in a 45-45-90 triangle, the hypotenuse's length is equal to the length of a leg multiplied by the square root of 2. This theorem stems directly from the Pythagorean theorem (a² + b² = c²), which applies to all right triangles. In a 45-45-90 triangle, since the legs are equal (let's call them 'a'), the Pythagorean theorem transforms into a² + a² = c², which simplifies to 2a² = c². Taking the square root of both sides gives us c = a√2, precisely the 45-45-90 triangle theorem. This theorem is not just a formula to memorize; it's a powerful shortcut. Instead of resorting to the full Pythagorean theorem each time, you can directly apply the 45-45-90 theorem to find the hypotenuse. This saves time and reduces the chance of errors, especially in complex problems. The theorem also highlights the elegant mathematical relationships inherent in geometric shapes. It demonstrates how specific triangle types have predictable side ratios, making calculations straightforward. Understanding the derivation of the theorem from the Pythagorean theorem provides a deeper appreciation for its validity and applicability. It connects seemingly disparate concepts, showcasing the interconnectedness of mathematical principles. So, whether you're solving textbook problems or tackling real-world design challenges, the 45-45-90 triangle theorem is your invaluable ally.

Solving for the Hypotenuse: A Step-by-Step Approach

Now, let's translate the 45-45-90 triangle theorem into a practical step-by-step method for finding the hypotenuse. The process is remarkably simple: 1. Identify the leg length. This is the starting point. In many problems, the leg length will be explicitly given. However, sometimes, you might need to deduce it from other information, such as the triangle's area or perimeter. 2. Multiply the leg length by √2. This is the direct application of the 45-45-90 triangle theorem. The result is the length of the hypotenuse. 3. Simplify the result, if necessary. The hypotenuse length might involve a radical (√2), which can sometimes be simplified further. For example, if the leg length is 5, the hypotenuse is 5√2, which is already in its simplest form. However, if the leg length were 5√2, the hypotenuse would be 5√2 * √2 = 5 * 2 = 10. Let's illustrate with our specific problem: each leg of a 45-45-90 triangle measures 14 cm. Following the steps: 1. Leg length = 14 cm. 2. Hypotenuse = 14 cm * √2 = 14√2 cm. 3. 14√2 is already in its simplest form. Therefore, the hypotenuse length is 14√2 cm. This step-by-step approach transforms a potentially abstract concept into a concrete, actionable method. It empowers you to confidently solve for the hypotenuse in any 45-45-90 triangle problem. Remember, the key is to understand the theorem and apply it systematically.

Applying the Concept: Example Problems and Solutions

To solidify your grasp of the 45-45-90 triangle theorem, let's examine a few example problems. These examples will showcase different scenarios and reinforce the step-by-step approach we discussed earlier. By working through these problems, you'll gain confidence in your ability to apply the theorem in various contexts. Furthermore, you'll develop a deeper understanding of the relationship between the legs and the hypotenuse in a 45-45-90 triangle. Let's consider a problem where the leg length is given as a radical expression. This will illustrate the simplification step more clearly. Example 1: A 45-45-90 triangle has legs of length 6√2 inches. What is the hypotenuse length? Solution: 1. Leg length = 6√2 inches. 2. Hypotenuse = 6√2 inches * √2 = 6 * 2 inches = 12 inches. This example demonstrates how multiplying by √2 twice effectively removes the radical. Now, let's explore a slightly more complex scenario where we need to determine the leg length before calculating the hypotenuse. Example 2: The area of a 45-45-90 triangle is 32 square centimeters. What is the hypotenuse length? Solution: 1. Recall that the area of a triangle is (1/2) * base * height. In a 45-45-90 triangle, the legs are the base and height. Let the leg length be 'a'. So, (1/2) * a * a = 32. 2. This simplifies to a² = 64. Taking the square root of both sides gives us a = 8 cm (leg length). 3. Hypotenuse = 8 cm * √2 = 8√2 cm. This example highlights the importance of connecting different geometric concepts. We used the area formula to find the leg length and then applied the 45-45-90 triangle theorem to find the hypotenuse. By working through these diverse examples, you'll hone your problem-solving skills and develop a versatile understanding of the 45-45-90 triangle.

The Solution to Our Problem: Finding the Hypotenuse with Legs of 14 cm

Now, let's directly address the problem posed in the title: each leg of a 45-45-90 triangle measures 14 cm. What is the length of the hypotenuse? We'll apply the step-by-step approach we've established to arrive at the solution. This will serve as a final demonstration of the 45-45-90 triangle theorem in action. By solving this specific problem, we'll not only find the answer but also reinforce the understanding we've built throughout this article. Let's break down the solution process: 1. Identify the leg length: The problem explicitly states that each leg measures 14 cm. So, the leg length is 14 cm. 2. Multiply the leg length by √2: According to the 45-45-90 triangle theorem, the hypotenuse is the leg length multiplied by √2. Therefore, the hypotenuse length is 14 cm * √2 = 14√2 cm. 3. Simplify the result, if necessary: The expression 14√2 is already in its simplest form. The square root of 2 is an irrational number and cannot be simplified further. Therefore, the hypotenuse length is 14√2 cm. Thus, the final answer is 14√2 cm. This concise solution demonstrates the power and efficiency of the 45-45-90 triangle theorem. By knowing the relationship between the legs and the hypotenuse, we can quickly and accurately determine the hypotenuse length without resorting to more complex calculations. This reinforces the value of understanding and applying geometric theorems. The solution also highlights the importance of recognizing and working with radical expressions, a fundamental skill in mathematics.

Conclusion: Mastering the 45-45-90 Triangle

In conclusion, the 45-45-90 triangle is a fundamental geometric shape with predictable side ratios. This article has explored its properties, with a particular focus on calculating the hypotenuse length. The 45-45-90 triangle theorem, which states that the hypotenuse is equal to the leg length multiplied by √2, is the key to solving these problems. We've outlined a clear, step-by-step approach for applying this theorem and worked through several examples to solidify your understanding. The ability to work confidently with 45-45-90 triangles is a valuable asset in various fields, including mathematics, engineering, and design. It provides a foundation for more advanced concepts and allows for efficient problem-solving. By mastering the principles discussed in this article, you've equipped yourself with a powerful tool for tackling geometrical challenges. Remember, the key is to understand the underlying concepts, not just memorize formulas. By connecting the 45-45-90 triangle theorem to the Pythagorean theorem, you gain a deeper appreciation for its validity and applicability. We encourage you to continue exploring the fascinating world of geometry and to apply your newfound knowledge to real-world situations. The 45-45-90 triangle is just one piece of a much larger puzzle, but it's a crucial piece that unlocks countless possibilities.