45-45-90 Triangle Hypotenuse 24 Inches Find The Leg Length
Introduction to 45-45-90 Triangles
In the realm of geometry, special right triangles hold a significant place due to their unique properties and applications. Among these, the 45-45-90 triangle stands out as a fundamental geometric shape with intriguing characteristics. This article delves into the intricacies of the 45-45-90 triangle, focusing on calculating the length of its legs when the hypotenuse is known. We will explore the underlying principles, relevant formulas, and step-by-step solutions, ensuring a comprehensive understanding of this fascinating triangle.
The 45-45-90 triangle, also known as an isosceles right triangle, is characterized by its angles: two angles measuring 45 degrees and one right angle (90 degrees). This unique configuration leads to a specific relationship between its sides. The two legs, opposite the 45-degree angles, are congruent, meaning they have the same length. The hypotenuse, opposite the right angle, is the longest side and is related to the legs by a constant factor. Understanding this relationship is crucial for solving problems involving 45-45-90 triangles.
One of the most critical aspects of the 45-45-90 triangle is the side length ratio. This ratio states that the lengths of the sides are in the proportion x : x : x√2, where x represents the length of each leg, and x√2 represents the length of the hypotenuse. This ratio is derived from the Pythagorean theorem, a fundamental concept in geometry that relates the sides of a right triangle. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). Applying this theorem to a 45-45-90 triangle allows us to establish the aforementioned ratio.
In practical applications, the 45-45-90 triangle appears in various fields, including architecture, engineering, and physics. For instance, it is used in the design of structures, the calculation of distances, and the analysis of forces. Its consistent side ratios make it a valuable tool for problem-solving in both theoretical and real-world scenarios. Mastering the properties of this triangle is essential for students and professionals alike, as it provides a foundation for understanding more complex geometric concepts and their applications. This article aims to equip you with the knowledge and skills necessary to confidently tackle problems involving 45-45-90 triangles, particularly those that involve finding the length of the legs given the hypotenuse.
Problem Statement: Finding the Leg Length
Our primary focus in this article is to address the specific problem: Given a 45-45-90 triangle with a hypotenuse of 24 inches, what is the length of one of the legs? This type of problem is common in geometry and requires a clear understanding of the relationships between the sides of a 45-45-90 triangle. Let's break down the problem and outline the steps required to solve it.
To solve this problem effectively, we need to understand the core concepts of a 45-45-90 triangle and how its sides relate to each other. As mentioned earlier, the side length ratio in a 45-45-90 triangle is x : x : x√2. Here, x represents the length of each leg, and x√2 represents the length of the hypotenuse. The problem provides us with the length of the hypotenuse (24 inches) and asks us to find the length of one leg, which we denote as x.
The key to solving this problem lies in setting up an equation that relates the given hypotenuse length to the unknown leg length. Since we know the hypotenuse is x√2, we can equate this expression to the given hypotenuse length (24 inches). This gives us the equation: x√2 = 24. The next step is to solve this equation for x, which will give us the length of one leg of the triangle. This involves algebraic manipulation to isolate x on one side of the equation.
Solving for x requires dividing both sides of the equation x√2 = 24 by √2. This gives us x = 24/√2. However, to express the answer in its simplest form, it is standard practice to rationalize the denominator. Rationalizing the denominator means eliminating the square root from the denominator of the fraction. To do this, we multiply both the numerator and the denominator of the fraction by √2. This process will give us a fraction with a rational denominator, making it easier to understand and compare with other values. The subsequent steps involve simplifying the resulting expression to arrive at the final answer for the length of one leg of the triangle.
This problem exemplifies a typical application of the 45-45-90 triangle properties. By understanding the side length ratio and applying basic algebraic principles, we can efficiently solve for unknown side lengths. The following sections will guide you through the detailed steps of solving this equation and provide a clear explanation of the reasoning behind each step, ensuring you grasp the underlying concepts and can apply them to similar problems in the future.
Step-by-Step Solution
Now, let's walk through the detailed solution to find the length of one leg of the 45-45-90 triangle when the hypotenuse is 24 inches. We will break down the process into clear, manageable steps, making it easy to follow and understand.
Step 1: Set up the equation. As discussed in the previous section, the relationship between the leg (x) and the hypotenuse (24 inches) in a 45-45-90 triangle is given by the equation: x√2 = 24. This equation is derived from the side length ratio of a 45-45-90 triangle, which is x : x : x√2. We are given the hypotenuse, which corresponds to x√2, and we need to find x, the length of one leg.
Step 2: Solve for x. To isolate x, we need to divide both sides of the equation by √2. This gives us: x = 24/√2. At this point, we have found the value of x, but it is not in its simplest form. The presence of a square root in the denominator is generally considered an unsimplified form. Therefore, the next step is to rationalize the denominator.
Step 3: Rationalize the denominator. To rationalize the denominator, we multiply both the numerator and the denominator of the fraction 24/√2 by √2. This process eliminates the square root from the denominator. Multiplying both the top and bottom by √2 gives us: x = (24 * √2) / (√2 * √2). Simplifying the denominator, we know that √2 * √2 = 2. So the equation becomes: x = (24√2) / 2.
Step 4: Simplify the fraction. Now we have x = (24√2) / 2. To simplify this fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 2. Dividing 24 by 2 gives us 12. Therefore, the simplified expression is: x = 12√2. This is the length of one leg of the 45-45-90 triangle, expressed in its simplest form.
Step 5: State the answer. The length of one leg of the 45-45-90 triangle is 12√2 inches. This is the final answer to the problem. We have successfully used the properties of 45-45-90 triangles and basic algebraic techniques to find the unknown leg length. Understanding each step in this process is crucial for solving similar problems in the future. The ability to manipulate equations and simplify expressions is a fundamental skill in mathematics and is particularly useful in geometry.
Answer and Explanation
Therefore, the length of one of the legs of the 45-45-90 triangle is 12√2 inches. Option B is the correct answer. This answer aligns with the principles and calculations we've discussed throughout this article.
To recap, we started with the given information: the hypotenuse of the 45-45-90 triangle measures 24 inches. Our goal was to find the length of one leg. We utilized the side length ratio of a 45-45-90 triangle (x : x : x√2) to set up an equation relating the leg length (x) to the hypotenuse (24 inches). This equation, x√2 = 24, was the foundation for our solution.
We then solved for x by dividing both sides of the equation by √2, resulting in x = 24/√2. This step is crucial in isolating the unknown variable and bringing us closer to the solution. However, as mentioned earlier, having a square root in the denominator is not considered the simplest form. To address this, we rationalized the denominator.
Rationalizing the denominator involves multiplying both the numerator and the denominator by the square root in the denominator. In this case, we multiplied both the numerator and the denominator by √2, resulting in x = (24√2) / (√2 * √2) = (24√2) / 2. This step eliminated the square root from the denominator, making the expression easier to simplify.
Finally, we simplified the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gave us the final answer: x = 12√2 inches. This answer represents the exact length of one leg of the 45-45-90 triangle. It is important to note that 12√2 is an exact value, whereas a decimal approximation would be an approximation. The exact value is often preferred in mathematical contexts.
The explanation highlights the importance of understanding the underlying principles of 45-45-90 triangles and how to apply algebraic techniques to solve geometric problems. The ability to set up equations, manipulate them, and simplify expressions is a core skill in mathematics. This example demonstrates a practical application of these skills in the context of a geometric problem. Furthermore, it emphasizes the significance of expressing answers in their simplest form, which often involves rationalizing denominators and reducing fractions.
Conclusion
In conclusion, understanding the properties of special right triangles, such as the 45-45-90 triangle, is fundamental to solving various geometric problems. This article has demonstrated a step-by-step approach to finding the length of a leg in a 45-45-90 triangle when the hypotenuse is known. By applying the side length ratio and algebraic manipulation, we can efficiently arrive at the correct answer.
The key takeaway from this discussion is the importance of the side length ratio x : x : x√2 in 45-45-90 triangles. This ratio provides a direct relationship between the legs and the hypotenuse, allowing us to set up equations and solve for unknown side lengths. The process of rationalizing the denominator, as demonstrated in this example, is a crucial technique for simplifying expressions and presenting answers in their most concise form. The ability to apply these concepts and techniques is essential for success in geometry and related fields.
Furthermore, this article emphasizes the broader applicability of mathematical principles in real-world scenarios. The 45-45-90 triangle, with its consistent properties, is used in various applications, including architecture, engineering, and design. The ability to solve problems involving this triangle is a valuable skill for professionals in these fields. Mastering the concepts discussed here will not only enhance your understanding of geometry but also equip you with the tools necessary to tackle practical problems in diverse contexts.
In summary, the problem of finding the leg length of a 45-45-90 triangle with a given hypotenuse highlights the interplay between geometric principles and algebraic techniques. By understanding the side length ratio, setting up appropriate equations, and applying algebraic manipulation, we can solve these problems effectively. The solution 12√2 inches not only answers the specific question but also reinforces the broader principles of problem-solving in mathematics and their relevance to practical applications. This knowledge provides a solid foundation for further exploration of geometry and its applications in the world around us.
