Hypotenuse Length Calculation For A 45-45-90 Triangle

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Understanding 45-45-90 Triangles

In the realm of geometry, the 45-45-90 triangle holds a special place due to its unique properties and consistent side ratios. This particular type of right triangle, also known as an isosceles right triangle, features two acute angles that each measure 45 degrees, and one right angle measuring 90 degrees. The distinct characteristic of this triangle lies in its side lengths, which follow a specific pattern that makes calculations relatively straightforward. Understanding the relationships between the sides of a 45-45-90 triangle is fundamental in various mathematical applications, including trigonometry, geometry, and even practical real-world scenarios. Before diving into the solution of our specific problem, it is crucial to grasp the underlying principle governing these triangles. The sides opposite the 45-degree angles, known as the legs, are always equal in length. The side opposite the 90-degree angle, the hypotenuse, has a length that is √2 times the length of each leg. This consistent ratio forms the basis for solving various problems involving 45-45-90 triangles. Recognizing this pattern allows us to quickly determine the length of any side if we know the length of another side. For instance, if we know the length of a leg, we can easily find the length of the hypotenuse by multiplying the leg length by √2. Conversely, if we know the length of the hypotenuse, we can find the length of each leg by dividing the hypotenuse length by √2. This inherent relationship simplifies calculations and makes 45-45-90 triangles a valuable tool in mathematical problem-solving. Moreover, the properties of 45-45-90 triangles extend beyond theoretical calculations. They find applications in various fields such as architecture, engineering, and construction. For example, architects might use the properties of these triangles to design structures with specific angles and proportions. Engineers might use them in calculating stress and strain in structural components. Even in everyday situations, understanding the principles of 45-45-90 triangles can be beneficial in tasks such as measuring distances, determining angles, and solving spatial reasoning problems. Thus, the study of these triangles not only enhances our mathematical knowledge but also equips us with practical skills applicable in various real-world scenarios.

Applying the 45-45-90 Triangle Theorem

The core of solving this problem lies in understanding and applying the 45-45-90 triangle theorem. As mentioned earlier, this theorem states that in a 45-45-90 triangle, the length of the hypotenuse is always √2 times the length of each leg. This fundamental relationship is the key to unlocking the solution. In our specific problem, we are given that each leg of the triangle measures 12 cm. We are asked to find the length of the hypotenuse. To do so, we simply apply the 45-45-90 triangle theorem. We multiply the length of each leg (12 cm) by √2 to obtain the length of the hypotenuse. Therefore, the length of the hypotenuse is 12√2 cm. This straightforward calculation demonstrates the power and elegance of the 45-45-90 triangle theorem. It allows us to quickly determine the unknown side length without resorting to more complex trigonometric calculations. The simplicity of the theorem makes it a valuable tool in various mathematical contexts. Whether you are solving geometry problems, calculating distances, or analyzing angles, the 45-45-90 triangle theorem provides a reliable and efficient method for finding the missing side lengths. Moreover, the theorem reinforces the importance of recognizing patterns in mathematics. By understanding the consistent relationship between the sides of a 45-45-90 triangle, we can easily solve a wide range of problems. This highlights the broader concept of mathematical problem-solving, which often involves identifying patterns, applying theorems, and using logical reasoning to arrive at a solution. In essence, the 45-45-90 triangle theorem serves as a cornerstone in the study of right triangles and provides a valuable foundation for more advanced mathematical concepts. Its simplicity and applicability make it an indispensable tool for students, mathematicians, and anyone dealing with geometric calculations.

Step-by-Step Solution

To illustrate the solution process, let's break down the steps involved in finding the hypotenuse length of the given 45-45-90 triangle:

  1. Identify the Triangle Type: Recognizing that the triangle is a 45-45-90 triangle is the first crucial step. This identification allows us to apply the specific properties and theorems associated with this type of triangle.
  2. State the Given Information: We are given that each leg of the triangle measures 12 cm. This information forms the basis for our calculation.
  3. Recall the 45-45-90 Triangle Theorem: The theorem states that the hypotenuse length is √2 times the length of each leg. This is the key formula we will use to solve the problem.
  4. Apply the Theorem: Multiply the length of each leg (12 cm) by √2: 12 cm * √2 = 12√2 cm.
  5. State the Answer: The length of the hypotenuse is 12√2 cm.

This step-by-step approach not only provides the correct answer but also emphasizes the importance of a structured problem-solving strategy. By breaking down the problem into manageable steps, we can avoid confusion and ensure accuracy. This method is applicable not only to this specific problem but also to a wide range of mathematical problems. It highlights the importance of understanding the underlying concepts, identifying relevant information, applying appropriate theorems, and systematically working towards a solution. Moreover, this approach fosters a deeper understanding of the problem-solving process itself. It encourages students to think critically, analyze information, and develop a logical approach to tackling mathematical challenges. By mastering this structured approach, students can build confidence in their problem-solving abilities and effectively tackle more complex mathematical problems in the future. In essence, the step-by-step solution provides a clear roadmap for solving problems involving 45-45-90 triangles and reinforces the importance of a systematic approach to mathematical problem-solving.

The Correct Answer and Why

The correct answer is D. 12√2 cm. This is because, as we established earlier, the hypotenuse of a 45-45-90 triangle is always √2 times the length of each leg. In this case, each leg measures 12 cm, so the hypotenuse measures 12 * √2 cm, which equals 12√2 cm. The other options are incorrect because they do not follow this established relationship. Option A, 6 cm, is incorrect because it represents half the length of a leg, not the hypotenuse. Option B, 6√2 cm, is incorrect because it represents the length of a leg divided by √2, which would be the length of a leg if we knew the hypotenuse. Option C, 12 cm, is incorrect because it is the length of a leg, not the hypotenuse. Understanding why the correct answer is correct is just as important as arriving at the correct answer itself. It demonstrates a thorough understanding of the underlying principles and concepts. In this case, understanding the 45-45-90 triangle theorem and its application is crucial for correctly solving the problem. Moreover, understanding why the incorrect answers are incorrect helps to solidify the understanding of the concept. It reinforces the specific relationship between the sides of a 45-45-90 triangle and prevents common mistakes. By analyzing the incorrect options, students can identify potential pitfalls and develop a deeper understanding of the nuances of the theorem. In essence, the process of understanding the correct answer and why the incorrect answers are incorrect promotes a more comprehensive and lasting understanding of the mathematical concepts involved. It encourages critical thinking, analytical skills, and a deeper appreciation for the logical reasoning behind mathematical solutions.

Conclusion: Mastering 45-45-90 Triangles

In conclusion, solving problems involving 45-45-90 triangles becomes significantly easier with a clear understanding of their unique properties and the application of the 45-45-90 triangle theorem. This theorem provides a direct relationship between the lengths of the legs and the hypotenuse, allowing for efficient calculations. The step-by-step solution outlined above demonstrates a structured approach to problem-solving, emphasizing the importance of identifying the triangle type, stating the given information, recalling the relevant theorem, applying the theorem, and stating the answer. By following this approach, students can confidently tackle problems involving 45-45-90 triangles and develop valuable problem-solving skills applicable to various mathematical contexts. Mastering the properties of 45-45-90 triangles is not just about solving specific problems; it's about developing a deeper understanding of geometric relationships and mathematical reasoning. This understanding extends beyond the classroom and finds applications in various real-world scenarios. Whether you are calculating distances, designing structures, or analyzing angles, the principles of 45-45-90 triangles provide a valuable tool for problem-solving. Moreover, the study of 45-45-90 triangles reinforces the importance of recognizing patterns in mathematics. By understanding the consistent relationship between the sides of these triangles, we can easily solve a wide range of problems and develop a more intuitive understanding of mathematical concepts. This highlights the broader goal of mathematics education, which is to equip students with the skills and knowledge necessary to solve problems, think critically, and apply mathematical principles in various contexts. In essence, mastering 45-45-90 triangles is a stepping stone towards a deeper understanding of geometry and mathematics as a whole.