Calculating Leg Length In A 45-45-90 Triangle With Hypotenuse Of 4 Cm

JU07/08/2025 08, 2025 by THE IDEN 70 views

The Length of a Leg in a 45-45-90 Triangle with a Hypotenuse of 4 cm

Introduction: Understanding 45-45-90 Triangles

In the realm of geometry, the 45-45-90 triangle stands out as a special right triangle with unique properties that simplify various calculations. Understanding the relationships between its sides is crucial for solving many geometric problems. A 45-45-90 triangle, also known as an isosceles right triangle, is characterized by two equal angles of 45 degrees and one right angle of 90 degrees. This unique angle configuration leads to a predictable and consistent ratio between the lengths of its sides, making it an essential concept in trigonometry and geometry. The sides opposite the 45-degree angles, known as the legs, are congruent, while the side opposite the 90-degree angle is the hypotenuse, which is the longest side of the triangle. The hypotenuse plays a pivotal role in determining the lengths of the legs, and understanding their relationship is fundamental to solving problems involving 45-45-90 triangles. This article delves into the specific problem of finding the length of a leg in a 45-45-90 triangle when the hypotenuse measures 4 cm. By applying the properties of these triangles and utilizing the Pythagorean theorem, we can accurately determine the leg length. This exploration will not only provide a solution to the given problem but also enhance your understanding of 45-45-90 triangles and their applications in various geometric contexts. The predictable nature of these triangles allows us to easily calculate unknown side lengths, making them a valuable tool in problem-solving and practical applications. Whether you are a student learning geometry or a professional applying these principles in real-world scenarios, a firm grasp of 45-45-90 triangles is invaluable.

The 45-45-90 Triangle Theorem: A Detailed Explanation

The cornerstone of solving problems involving 45-45-90 triangles lies in understanding the 45-45-90 Triangle Theorem. This theorem elucidates the relationship between the lengths of the legs and the hypotenuse. Specifically, in a 45-45-90 triangle, the lengths of the legs are equal, and the length of the hypotenuse is $\sqrt{2}$ times the length of a leg. This relationship is derived from the Pythagorean theorem, which states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). For a 45-45-90 triangle, if we denote the length of each leg as x, then the Pythagorean theorem can be expressed as: $x^2 + x^2 = hypotenuse^2$ . Simplifying this equation, we get $2x^2 = hypotenuse^2$ . Taking the square root of both sides, we find that the hypotenuse is equal to $x\sqrt{2}$ . This formula is the essence of the 45-45-90 Triangle Theorem. It provides a direct and efficient method for determining the length of any side if the length of another side is known. For instance, if we know the length of a leg, we can multiply it by $\sqrt{2}$ to find the length of the hypotenuse. Conversely, if we know the length of the hypotenuse, we can divide it by $\sqrt{2}$ to find the length of each leg. This theorem not only simplifies calculations but also provides a clear conceptual understanding of the geometric relationships within a 45-45-90 triangle. It is a fundamental tool in geometry and trigonometry, enabling students and professionals to solve a wide range of problems involving these special right triangles. Grasping this theorem is essential for anyone seeking to master geometric problem-solving and apply these concepts in practical scenarios.

Solving the Problem: Finding the Leg Length

Now, let's apply the 45-45-90 Triangle Theorem to solve the specific problem at hand. We are given that the hypotenuse of a 45-45-90 triangle measures 4 cm, and our goal is to find the length of one leg. According to the theorem, the relationship between the hypotenuse (H) and the leg (L) in a 45-45-90 triangle is given by the formula: $H = L\sqrt{2}$ . We can rearrange this formula to solve for the length of the leg (L): $L = \frac{H}{\sqrt{2}}$ . Substituting the given value of the hypotenuse (H = 4 cm) into the formula, we get: $L = \frac{4}{\sqrt{2}}$ . To simplify this expression, we need to rationalize the denominator. Rationalizing the denominator involves eliminating the square root from the denominator of a fraction. We can achieve this by multiplying both the numerator and the denominator by $\sqrt{2}$ : $L = \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$ . This simplifies to: $L = \frac{4\sqrt{2}}{2}$ . Further simplification yields: $L = 2\sqrt{2}$ cm. Therefore, the length of one leg of the 45-45-90 triangle is $2\sqrt{2}$ cm. This solution demonstrates the practical application of the 45-45-90 Triangle Theorem. By understanding the relationship between the sides of this special triangle and applying basic algebraic manipulation, we can efficiently determine unknown side lengths. This approach is not only applicable to this specific problem but also serves as a general method for solving similar problems involving 45-45-90 triangles. The ability to quickly and accurately solve these types of problems is a valuable skill in geometry and related fields.

Step-by-Step Solution: A Detailed Walkthrough

To ensure clarity and understanding, let's provide a step-by-step walkthrough of the solution to find the leg length of the 45-45-90 triangle with a hypotenuse of 4 cm:

Identify the Given Information: We are given that the hypotenuse (H) of the 45-45-90 triangle is 4 cm.
Recall the 45-45-90 Triangle Theorem: The theorem states that in a 45-45-90 triangle, the relationship between the hypotenuse (H) and the leg (L) is $H = L\sqrt{2}$ .
Write the Formula to Find the Leg Length: We need to rearrange the formula to solve for L (the leg length): $L = \frac{H}{\sqrt{2}}$ .
Substitute the Given Value: Substitute the value of the hypotenuse (H = 4 cm) into the formula: $L = \frac{4}{\sqrt{2}}$ .
Rationalize the Denominator: To eliminate the square root from the denominator, multiply both the numerator and the denominator by $\sqrt{2}$ : $L = \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$ .
Simplify the Expression: Multiply the numerators and the denominators: $L = \frac{4\sqrt{2}}{2}$ .
Reduce the Fraction: Divide both the numerator and the denominator by their greatest common divisor, which is 2: $L = 2\sqrt{2}$ cm.
State the Answer: The length of one leg of the 45-45-90 triangle is $2\sqrt{2}$ cm.

This step-by-step solution breaks down the problem into manageable steps, making it easier to follow and understand. Each step is clearly explained, ensuring that the logic and reasoning behind the solution are transparent. This approach is beneficial for learners who are new to 45-45-90 triangles and provides a solid foundation for solving more complex problems in the future. By following these steps, you can confidently solve similar problems involving 45-45-90 triangles and other special right triangles.

Conclusion: Key Takeaways and Applications

In conclusion, this article has explored the properties of 45-45-90 triangles and demonstrated how to find the length of a leg when the hypotenuse is known. The key takeaway is the 45-45-90 Triangle Theorem, which establishes the relationship between the lengths of the legs and the hypotenuse. Specifically, in a 45-45-90 triangle, the legs are equal in length, and the hypotenuse is $\sqrt{2}$ times the length of a leg. This relationship allows us to efficiently solve for unknown side lengths using the formula $L = \frac{H}{\sqrt{2}}$ , where L is the length of a leg and H is the length of the hypotenuse. By applying this formula and rationalizing the denominator when necessary, we can accurately determine the lengths of the sides in a 45-45-90 triangle. In the given problem, where the hypotenuse measures 4 cm, we found that the length of one leg is $2\sqrt{2}$ cm. This solution highlights the practical application of the 45-45-90 Triangle Theorem and demonstrates a step-by-step approach to solving such problems. The concepts and methods discussed in this article have wide-ranging applications in various fields, including architecture, engineering, construction, and navigation. For example, architects and engineers often use the properties of 45-45-90 triangles to design structures and ensure precise measurements. In construction, these principles are applied in tasks such as setting foundations and aligning walls. Navigators utilize these triangles to calculate distances and bearings. Moreover, understanding 45-45-90 triangles is essential for success in higher-level mathematics courses, such as trigonometry and calculus. The ability to quickly and accurately solve problems involving these triangles is a valuable skill that can enhance your problem-solving capabilities in various contexts. Therefore, mastering the 45-45-90 Triangle Theorem and its applications is a worthwhile endeavor for students, professionals, and anyone interested in geometry and its practical uses.