Finding The Leg Length Of A 45-45-90 Triangle

In the realm of geometry, special right triangles hold a position of particular significance, offering a set of readily applicable rules and relationships that streamline problem-solving. Among these, the 45-45-90 triangle stands out as a classic example, distinguished by its unique angle measures and side length ratios. This article delves into the fascinating properties of the 45-45-90 triangle, focusing on how to determine the length of its legs when the hypotenuse is known. Specifically, we will address the problem of finding the leg length when the hypotenuse is 10 units long. Understanding these relationships is fundamental for anyone studying geometry, trigonometry, or related fields. This exploration will not only provide a step-by-step solution to the problem but also offer insights into the underlying principles that make these calculations possible. By mastering the concepts associated with 45-45-90 triangles, students and enthusiasts alike can enhance their problem-solving skills and deepen their appreciation for the elegance of geometric relationships.

Before we tackle the specific problem, let's establish a solid foundation by understanding the characteristics of a 45-45-90 triangle. This type of triangle is a special case of a right triangle, which means it has one 90-degree angle. What makes it special is that the other two angles are both 45 degrees. This unique angle combination leads to a fascinating relationship between the sides. In a 45-45-90 triangle, the two legs (the sides that form the right angle) are always congruent, meaning they have the same length. This is a direct consequence of the fact that the angles opposite these sides are equal. The hypotenuse, which is the side opposite the right angle, is the longest side of the triangle. The sides of a 45-45-90 triangle are in a special ratio. If we denote the length of each leg as x, then the length of the hypotenuse is x√2. This ratio is derived from the Pythagorean theorem and is a cornerstone for solving problems involving these triangles. Remembering this ratio is crucial because it provides a shortcut for finding side lengths without having to use the Pythagorean theorem directly every time. The 45-45-90 triangle is not just a theoretical construct; it appears in various real-world applications, from architecture and engineering to everyday geometry problems. Its predictable side ratios make it a valuable tool for calculations and designs. Grasping the properties of the 45-45-90 triangle is essential for anyone delving into trigonometry and advanced geometric concepts. The simplicity and elegance of its relationships offer a gateway to understanding more complex geometric figures and problem-solving techniques.

Now, let's clearly state the problem we aim to solve: The hypotenuse of a 45-45-90 triangle has a length of 10 units. What is the length of one of its legs? This problem is a classic application of the properties of 45-45-90 triangles. It requires us to use the known relationship between the hypotenuse and the legs to find the unknown leg length. The hypotenuse, being the longest side, is given, and we need to determine the length of either of the two congruent legs. This problem underscores the importance of understanding the special ratio that governs the sides of a 45-45-90 triangle. While the Pythagorean theorem could technically be used to solve this, the direct application of the ratio provides a more efficient and elegant solution. This type of problem is common in geometry courses and standardized tests, making it essential for students to master the underlying concepts. By solving this problem, we'll not only find the numerical answer but also reinforce our understanding of how 45-45-90 triangle properties can be applied in practical situations. The clarity of the problem statement is crucial because it sets the stage for a focused and strategic approach to the solution. Knowing exactly what we're trying to find allows us to select the most appropriate method and avoid unnecessary steps.

To solve this problem, we will leverage the special ratio inherent in 45-45-90 triangles. As we discussed earlier, if the legs of the triangle have a length of x, then the hypotenuse has a length of x√2. In our case, we know the hypotenuse is 10 units long. Therefore, we can set up the equation x√2 = 10. This equation directly relates the unknown leg length (x) to the known hypotenuse length. The next step is to solve for x. To isolate x, we divide both sides of the equation by √2, which gives us x = 10 / √2. While this is a valid answer, it's common practice to rationalize the denominator, meaning we want to eliminate the square root from the denominator. To do this, we multiply both the numerator and the denominator by √2. This gives us x = (10√2) / (√2 * √2), which simplifies to x = (10√2) / 2. Now, we can further simplify by dividing both the numerator and the denominator by 2, resulting in x = 5√2. Therefore, the length of each leg of the 45-45-90 triangle is 5√2 units. This solution demonstrates the power of understanding and applying the special ratios of 45-45-90 triangles. By using this ratio, we were able to efficiently solve the problem without resorting to more complex methods. This approach not only saves time but also showcases the elegance and efficiency of geometric principles. The ability to quickly recognize and apply these ratios is a valuable skill in geometry and related fields.

Let's break down the calculation into a clear, step-by-step process for better understanding.

Step 1: Identify the Relationship: Recall the 45-45-90 triangle ratio: if the legs are of length x, the hypotenuse is x√2. This is the foundational principle we'll use.

Step 2: Set Up the Equation: We know the hypotenuse is 10 units. So, we set up the equation x√2 = 10. This equation translates the geometric relationship into an algebraic one.

Step 3: Isolate x: To find the leg length x, we divide both sides of the equation by √2: x = 10 / √2. This isolates x but leaves us with a square root in the denominator.

Step 4: Rationalize the Denominator: To remove the square root from the denominator, we multiply both the numerator and the denominator by √2: x = (10√2) / (√2 * √2). This is a standard algebraic technique to simplify expressions.

Step 5: Simplify: Simplify the expression: x = (10√2) / 2. We've now eliminated the square root from the denominator and have a clearer fraction.

Step 6: Further Simplification: Divide both the numerator and the denominator by 2: x = 5√2. This gives us the final, simplified answer.

Step 7: State the Answer: Therefore, the length of one leg of the 45-45-90 triangle is 5√2 units. This step-by-step approach clarifies the logical progression of the solution, making it easier to follow and understand. Each step is a deliberate action based on mathematical principles, ensuring accuracy and efficiency. This detailed breakdown is particularly helpful for students learning these concepts, as it provides a clear roadmap for solving similar problems. By following these steps, anyone can confidently tackle problems involving 45-45-90 triangles and their unique properties.

Based on our calculations, the length of one leg of the 45-45-90 triangle is 5√2 units. Therefore, the correct answer is C. 5√2 units. This result aligns perfectly with the principles we've discussed and the step-by-step solution we've outlined. To reiterate, we started with the understanding that in a 45-45-90 triangle, the hypotenuse is √2 times the length of a leg. Knowing the hypotenuse was 10 units, we set up the equation x√2 = 10, where x represents the leg length. We then solved for x by dividing both sides by √2, rationalizing the denominator, and simplifying the expression. This process led us directly to the answer of 5√2 units. Understanding why the other options are incorrect is also important. Option A, 10√2 units, would be the length of the hypotenuse if the legs were 10 units long, not the other way around. Option B, 10 units, would be the leg length only if the triangle were an isosceles right triangle with legs equal to the hypotenuse, which is not possible. Option D, 5 units, would be the leg length if we had incorrectly divided the hypotenuse by 2 instead of √2. The correct answer, 5√2 units, is the only one that satisfies the 45-45-90 triangle ratio and the given hypotenuse length. This comprehensive explanation reinforces the accuracy of our solution and provides a clear understanding of why the other choices are incorrect.

In conclusion, determining the leg length of a 45-45-90 triangle when the hypotenuse is known involves a straightforward application of the triangle's special properties. We have demonstrated that if the hypotenuse is 10 units, the length of each leg is 5√2 units. This solution is derived from the fundamental ratio that governs 45-45-90 triangles: the hypotenuse is always √2 times the length of a leg. By setting up the equation x√2 = 10, we were able to solve for x, the leg length, through a series of algebraic manipulations, including rationalizing the denominator. The step-by-step approach we outlined provides a clear and concise method for tackling similar problems. This understanding is crucial for anyone studying geometry, trigonometry, or related fields. The 45-45-90 triangle is a cornerstone concept, and mastering its properties opens the door to more complex geometric problem-solving. Furthermore, the ability to quickly and accurately solve such problems is valuable not only in academic settings but also in practical applications where geometric principles are relevant. The elegance and efficiency of the 45-45-90 triangle ratio highlight the beauty of mathematical relationships and their power in simplifying calculations. This exploration reinforces the importance of understanding special right triangles and their unique characteristics.