Finding The Leg Length Of A 45-45-90 Triangle With 18 Cm Hypotenuse
In the realm of geometry, the 45-45-90 triangle holds a special place due to its unique properties and straightforward relationships between its sides. These triangles, also known as isosceles right triangles, possess two equal angles of 45 degrees, one right angle (90 degrees), and two congruent legs. Understanding the proportions within these triangles is crucial for solving various geometric problems. This article delves into a specific problem involving a 45-45-90 triangle where the hypotenuse measures 18 cm and aims to determine the length of one of its legs. We will explore the underlying principles, the mathematical reasoning, and the step-by-step solution to this problem, providing a comprehensive understanding of the concepts involved.
Understanding 45-45-90 Triangles
Before diving into the problem, it's essential to grasp the fundamental properties of 45-45-90 triangles. These triangles derive their unique characteristics from their angle measures, which dictate the ratios between their sides. The sides of a 45-45-90 triangle are always in a specific ratio: 1 : 1 : √2. This ratio represents the relationship between the lengths of the two legs and the hypotenuse. If we denote the length of each leg as 'x', then the hypotenuse will have a length of 'x√2'. This relationship is a direct consequence of the Pythagorean theorem and the angle properties of these triangles. The Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, plays a pivotal role in understanding this ratio. In a 45-45-90 triangle, since the two legs are equal, the theorem simplifies to: hypotenuse² = leg² + leg² = 2 * leg². Taking the square root of both sides and considering the side ratios leads to the fundamental 1 : 1 : √2 proportion. This inherent proportion makes 45-45-90 triangles predictable and easy to work with in various geometric scenarios. Recognizing this ratio allows for quick calculations and solutions when dealing with these types of triangles. Furthermore, the isosceles nature of the triangle, with its two equal legs and two equal angles, contributes to its symmetrical properties, making it a fundamental shape in geometry and trigonometry.
Problem Statement: Finding the Leg Length
The problem at hand presents a 45-45-90 triangle with a hypotenuse measuring 18 cm. The objective is to determine the length of one of the legs of the triangle. This problem is a classic application of the 45-45-90 triangle properties, requiring the use of the side ratio we discussed earlier. To solve this, we need to work backward from the known hypotenuse length to find the length of the leg. We know that the ratio of the hypotenuse to a leg in a 45-45-90 triangle is √2 : 1. This relationship stems from the fundamental proportion of the sides in these triangles. By understanding this ratio, we can set up an equation that relates the given hypotenuse length to the unknown leg length. This approach allows us to apply algebraic principles to solve for the missing side. The problem tests not only the understanding of the 45-45-90 triangle properties but also the ability to apply these properties in a practical context. It requires a clear understanding of how the sides of the triangle relate to each other and the ability to use this relationship to find unknown lengths. The problem is a good example of how geometric principles can be used to solve real-world problems involving shapes and measurements.
Solution: Step-by-Step Calculation
To solve for the leg length, let's denote the length of one leg as 'x'. As established earlier, the hypotenuse of a 45-45-90 triangle is √2 times the length of a leg. Therefore, we can express the relationship as: Hypotenuse = x√2. In this problem, the hypotenuse is given as 18 cm. So, we can set up the equation: 18 = x√2. To isolate 'x' and find the leg length, we need to divide both sides of the equation by √2: x = 18 / √2. To rationalize the denominator, we multiply both the numerator and the denominator by √2: x = (18 * √2) / (√2 * √2). This simplifies to: x = (18√2) / 2. Now, we can simplify the fraction by dividing both the numerator and the denominator by 2: x = 9√2. Therefore, the length of one leg of the triangle is 9√2 cm. This step-by-step solution demonstrates how the fundamental properties of 45-45-90 triangles and algebraic manipulation can be combined to solve geometric problems. The process of rationalizing the denominator is a crucial step to ensure the answer is in its simplest form. The final result, 9√2 cm, represents the exact length of the leg, highlighting the importance of understanding and applying mathematical principles accurately.
Answer and Explanation
Based on our calculations, the length of one leg of the 45-45-90 triangle is 9√2 cm, which corresponds to option B. This answer is derived from the inherent properties of 45-45-90 triangles and the application of basic algebraic principles. The key to solving this problem lies in understanding the relationship between the sides of a 45-45-90 triangle, specifically the ratio of 1 : 1 : √2 between the legs and the hypotenuse. By recognizing this ratio and setting up the appropriate equation, we were able to solve for the unknown leg length. The process of rationalizing the denominator further simplified the result, providing a clear and concise answer. This solution highlights the importance of memorizing and understanding the special triangle ratios, as they can significantly simplify problem-solving in geometry. The correct answer, 9√2 cm, not only solves the problem but also reinforces the understanding of the mathematical principles underlying 45-45-90 triangles. The other options presented, 9 cm, 18 cm, and 18√2 cm, are incorrect because they do not accurately reflect the side ratios in a 45-45-90 triangle with a hypotenuse of 18 cm. This emphasizes the need for a thorough understanding of the relationships between the sides in these special triangles.
Why Other Options Are Incorrect
To further solidify the understanding of the solution, let's examine why the other options are incorrect. Option A, 9 cm, is incorrect because it doesn't account for the √2 factor in the relationship between the leg and the hypotenuse in a 45-45-90 triangle. If a leg were 9 cm, the hypotenuse would be 9√2 cm, not 18 cm. Option C, 18 cm, is simply the length of the hypotenuse, not the leg. This option misunderstands the question and the relationship between the sides. Option D, 18√2 cm, would be the length of the hypotenuse if the legs were 18 cm each. This value is larger than the given hypotenuse, indicating a misunderstanding of the side ratios. Each incorrect option represents a different misunderstanding of the triangle's properties. Option A overlooks the √2 factor, Option C confuses the leg and hypotenuse, and Option D inverts the relationship. Understanding why these options are wrong reinforces the correct methodology and the importance of applying the 45-45-90 triangle ratios accurately. By analyzing the errors in these options, students can develop a deeper understanding of the underlying mathematical principles and avoid similar mistakes in the future. This detailed explanation of why the other options are incorrect serves as a valuable learning tool, promoting a comprehensive grasp of the subject matter.
Conclusion
In conclusion, the length of one leg of the 45-45-90 triangle with a hypotenuse of 18 cm is 9√2 cm. This solution is obtained by applying the fundamental properties of 45-45-90 triangles, which dictate a specific ratio between the lengths of the legs and the hypotenuse. The ratio 1 : 1 : √2 is crucial for solving problems involving these triangles. By understanding this ratio and using basic algebraic principles, we can efficiently determine unknown side lengths. This problem serves as a valuable exercise in applying geometric concepts to practical scenarios. The step-by-step solution demonstrates the importance of a clear and methodical approach to problem-solving in mathematics. Furthermore, the analysis of why other options are incorrect reinforces the understanding of the underlying principles and common pitfalls to avoid. Mastering the properties of special triangles, such as the 45-45-90 triangle, is essential for success in geometry and related fields. This knowledge enables quick and accurate calculations, which are invaluable in various applications. The ability to apply these concepts effectively showcases a strong understanding of geometric principles and mathematical reasoning.
