Finding The Leg Length Of A 45-45-90 Triangle With Hypotenuse 128 Cm

In the realm of geometry, the 45-45-90 triangle stands as a special right triangle, celebrated for its unique properties and elegant relationships between its sides. This article delves into the specifics of a 45-45-90 triangle where the hypotenuse measures 128 cm, aiming to determine the length of one of its legs. We will explore the fundamental characteristics of such triangles and employ the 45-45-90 triangle theorem to arrive at the solution. By understanding the ratios inherent in these triangles, we can efficiently solve for unknown side lengths, making this a crucial concept for students and enthusiasts of mathematics.

Understanding the 45-45-90 Triangle

The 45-45-90 triangle, also known as an isosceles right triangle, possesses two equal angles of 45 degrees and one right angle of 90 degrees. This unique angular configuration leads to a specific relationship between the lengths of its sides. The two legs, which are opposite the 45-degree angles, are congruent, meaning they have the same length. The hypotenuse, which is the side opposite the 90-degree angle, is longer than either leg. The beauty of the 45-45-90 triangle lies in the consistent ratio between its sides, a property that simplifies calculations and problem-solving in geometry. Knowing this ratio allows us to quickly determine unknown side lengths when one side is given, making it a powerful tool in various mathematical contexts. Understanding the properties of this triangle not only aids in solving geometric problems but also enhances our appreciation for the elegance and precision of mathematical relationships.

The defining feature of a 45-45-90 triangle is the ratio of its sides. If we denote the length of each leg as x, then the length of the hypotenuse is x√2. This ratio stems from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In a 45-45-90 triangle, this translates to x² + x² = h², where h is the length of the hypotenuse. Simplifying this equation, we get 2x² = h², and taking the square root of both sides gives us h = x√2. This fundamental relationship is the cornerstone of solving problems involving 45-45-90 triangles. Recognizing and applying this ratio is key to efficiently finding unknown side lengths, whether you're working on geometry problems or tackling real-world applications that involve triangular shapes. The consistency of this ratio makes the 45-45-90 triangle a predictable and valuable geometric figure.

The significance of the 45-45-90 triangle extends beyond textbook problems; it appears frequently in various real-world applications. Architecture, engineering, and construction often utilize this triangle in designs and calculations due to its inherent stability and predictable proportions. For instance, the diagonal bracing in structures often forms 45-45-90 triangles, providing crucial support and reinforcement. In navigation, understanding the properties of this triangle can aid in calculating distances and bearings. Furthermore, in computer graphics and game development, 45-45-90 triangles are used in creating and manipulating shapes and objects. The prevalence of this triangle in practical scenarios underscores its importance in mathematics education and its relevance to numerous fields. By mastering the concepts related to 45-45-90 triangles, students not only enhance their geometric problem-solving skills but also gain a valuable tool for analyzing and understanding the world around them.

Applying the 45-45-90 Triangle Theorem

The 45-45-90 triangle theorem is the principle that governs the relationship between the sides of a 45-45-90 triangle. As established earlier, this theorem states that if the length of each leg is x, then the length of the hypotenuse is x√2. Conversely, if we know the length of the hypotenuse, we can find the length of each leg by dividing the hypotenuse by √2. This theorem is a direct application of the Pythagorean theorem, tailored specifically to the unique characteristics of the 45-45-90 triangle. It provides a straightforward method for calculating unknown side lengths, eliminating the need for more complex trigonometric functions. The 45-45-90 triangle theorem is a powerful tool for quick and accurate problem-solving in geometry. Mastering this theorem is essential for anyone studying trigonometry or geometry, as it simplifies many calculations involving right triangles with 45-degree angles. The elegance and efficiency of this theorem highlight the beauty and practicality of mathematical principles.

In the context of our problem, we are given that the hypotenuse of the 45-45-90 triangle measures 128 cm. To find the length of one leg, we need to apply the 45-45-90 triangle theorem in reverse. We know that the hypotenuse is x√2, where x is the length of each leg. Therefore, we can set up the equation x√2 = 128 cm. To solve for x, we divide both sides of the equation by √2, which gives us x = 128 cm / √2. To rationalize the denominator, we multiply both the numerator and the denominator by √2, resulting in x = (128 cm * √2) / 2. Simplifying this expression, we get x = 64√2 cm. This calculation demonstrates the practical application of the 45-45-90 triangle theorem in determining unknown side lengths. By following this step-by-step process, we can confidently find the length of a leg in a 45-45-90 triangle when the hypotenuse is known.

To further illustrate the application of the 45-45-90 triangle theorem, let's consider a few more examples. Imagine a square with sides of length 10 cm. If we draw a diagonal across the square, we create two 45-45-90 triangles. The diagonal of the square becomes the hypotenuse of each triangle. Using the theorem, we can calculate the length of the diagonal as 10√2 cm. Another example could involve a ramp that forms a 45-degree angle with the ground. If the horizontal distance covered by the ramp is 5 meters, we can use the 45-45-90 triangle theorem to find the length of the ramp (the hypotenuse), which would be 5√2 meters. These examples highlight the versatility of the theorem in solving various geometric problems. By practicing with different scenarios, one can gain a deeper understanding of how to apply the theorem effectively and efficiently. The key is to recognize the 45-45-90 triangle pattern and then apply the appropriate formula to find the unknown side lengths.

Solution: Finding the Leg Length

As we have established, the hypotenuse of the given 45-45-90 triangle measures 128 cm. Applying the 45-45-90 triangle theorem, we know that the length of each leg is the hypotenuse divided by √2. Therefore, the length of one leg is 128 cm / √2. To rationalize the denominator, we multiply both the numerator and the denominator by √2, which gives us (128 cm * √2) / 2. Simplifying this, we find that the length of one leg is 64√2 cm. This result matches option B in the given choices.

Therefore, the correct answer to the question is B. 64√2 cm. This solution demonstrates the direct application of the 45-45-90 triangle theorem in solving for an unknown side length. By understanding the relationship between the sides of a 45-45-90 triangle, we can efficiently calculate the length of a leg when the hypotenuse is known. This skill is valuable in various mathematical contexts, including geometry, trigonometry, and real-world applications involving triangular shapes.

In conclusion, understanding and applying the 45-45-90 triangle theorem is crucial for solving problems involving these special right triangles. The consistent ratio between the sides allows for quick and accurate calculations. In the case of a 45-45-90 triangle with a hypotenuse of 128 cm, the length of one leg is 64√2 cm. This knowledge not only helps in academic settings but also provides a foundation for practical applications in various fields that utilize geometric principles.